Array Processing for Seismic Surface Waves

Diss. ETH No. 21291

A dissertation submitted to ETH Zurich for the degree of Doctor of Sciences

presented by Stefano Maranò Laurea Specialistica in Ingegneria delle Telecomunicazioni, Università degli Studi di Trento born on December 23, 1983 citizen of Italy

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

2013 ii

Abstract

The analysis of seismic surface waves plays a major role in the understanding of geological and geophysical features of the subsoil. Indeed seismic wave attributes such as velocity of propagation or wave polarization reflect the properties of the materials in which the wave is propagating. The analysis of properties of surface waves allows geophysicists to gain insight into the structure of the subsoil avoiding more expensive invasive techniques (e.g., borehole techniques). A myriad of applications benefit from the knowledge about the subsoil gained through seismic surveys. Microzonation studies are an important application of the analysis of surface waves with direct impact on damage mitigation and earthquake preparedness. This thesis aims at improving signal processing techniques for the analysis of surface waves in different directions. In particular, the main goal is to deliver accurate estimates of the geophysical parameters of interest. The availability of improved estimates of the quantities of interest will provide better constraints for the geophysical inversion and thus enabling us to obtain an improved structural earth model. For a rigorous treatment of the estimation of wavefield parameters we rely on tools from statistical signal processing. Wavefield parameters are estimated using the maximum likelihood (ML) method. A computationally efficient implementation of such an estimator is obtained by modelling seismic surface waves with a factor graph, a particular type of probabilistic graphical model. A theoretical bound on estimation accuracy, the Cramér-Rao bound (CRB), enables us to quantify the sources of uncertainty and provides a benchmark for evaluating estimation algorithms. One main contribution of this work is the development of a method for the analysis of seismic surface waves. The method is versatile enough to model different types of waves and to handle measurements of different type. All the wavefield parameters of Love waves and Rayleigh waves, together with all the measurements, are jointly modelled within the proposed framework. The method ensures an optimal usage of the available measurements according to the ML criterion. The method also deals with the simultaneous presence of multiple waves, possibly of different type. The proposed algorithm decomposes the wavefield by gradually increasing the number of waves modelled and iteratively refining estimates of the parameters of each wave. Sensors with different noise level are also accounted for and the estimation accounts for the possible different quality of the measurements. Performance is assessed on field measurements of ambient vibrations from sensor arrays. It is shown how the proposed method outperforms methods in the literature in different ways, namely: Rayleigh wave ellipticity is retrieved with increased accuracy, the retrograde/prograde particle motion of the Rayleigh wave is retrieved for the first time, and the simultaneous presence of multiple waves is considered. It is also shown that the implementation of the proposed method exhibits, for a sufficiently large signal-to-noise ratio (SNR), the smallest achievable mean-squared estimation error (MSEE) indicated by the CRB. The joint processing of translational and rotational motions is tested on recordings from controlled explosions. We show the retrieval of Love and Rayleigh wave parameters in several settings not considered in the literature, both in the case of a single six-component sensor and the case iii of an array of three and six-component sensors. Analytic expressions of the CRB of each parameter of geophysical interest are derived. These expressions allow us to quantify and understand the sources of uncertainty limiting the estimation accuracy of the wavefield parameters. The statistical models for Love and Rayleigh waves relying on translational measurements, rotational measurements, and both translational and rotational measurements are considered. The impact of array geometry on the estimation of parameters of Love and Rayleigh waves is also investigated. It is explained in detail how the array geometry affects the MSEE of parameters of interest, such as 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 Love and Rayleigh waves. Several computational approaches to minimize the proposed cost function are presented and compared. Finally, numerical experiments verify the effectiveness of the proposed cost function and resulting array geometry designs, leading to greatly improved estimation performance in comparison to arbitrary array geometries, both at low and high SNR levels. iv

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 seismischen 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 seismische Messungen gewonnen werden, kommen einer Vielzahl von Anwendungen zugute. Mikrozonierungen stellen eine wichtige Anwendung in der Ana- lyse von Oberflächenwellen dar, und haben einen direkten Einfluss auf die Vorbereitung auf Erdbeben und die Begrenzung von Schäden. Diese Dissertation zielt darauf ab, Signalverarbeitungstechniken zur Analyse 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 Randbedingungen für die geophysikalische Inversion bereitstellen und erlauben, verbesserte strukturelle Modelle der Erde zu erhalten. Für eine rigorose Behandlung der Schätzung der Wellenfeldparameter setzen wir auf Werkzeuge aus der statistischen Signalverarbeitung. Wel- lenfeldparameter werden mit Hilfe der Maximum-Likelihood-Methode geschätzt. Eine rechnerisch effiziente Umsetzung solcher Schätzer wird durch 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 Bewertung der Schätzalgorithmen dar. Ein Hauptbeitrag dieser Arbeit ist die Entwicklung eines Verfahrens zur Analyse von seismischen Oberflächenwellen. Das Verfahren ist flexi- bel genug, um verschiedene Arten von Wellen zu modellieren und unterschiedliche Messmethoden zu handhaben. Alle Wellenfeldparameter der Love- und Rayleighwellen zusammen mit all den Messungen werden ge- meinsam modelliert. Das Verfahren gewährleistet eine optimale Nutzung der verfügbaren Messungen nach dem Maximum-Likelihood Kriterium. Das Verfahren behandelt auch die gleichzeitige Anwesenheit von mehreren Wellen, möglicherweise von unterschiedlichen Wellentypen. Der vorgestellte Algorithmus zerlegt das Wellenfeld durch allmähliches Er- höhen der Anzahl modellierter Wellen und verfeinert die Schätzung der Parameter der einzelnen Wellen iterativ. Sensoren mit unterschiedlichem Geräuschpegel 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 wird gezeigt, dass das vorgeschlagene Verfahren den Methoden aus der Li- teratur in unterschiedlichen Aspekten überlegen ist. Insbesondere wird die Elliptizität der Rayleighwelle mit verbesserter Auflösung wiederge- geben, die retrograde/prograde Partikelbewegung wird zum ersten Mal hergeleitet, und die gleichzeitige Anwesenheit von mehreren Wellen wird betrachtet. v

Es wird auch gezeigt, dass die Umsetzung der vorgestellten Methode für ein ausreichend grosses Signal-Rausch-Verhältnis (SRV) den kleinsten erreichbaren 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 drei und sechs Komponenten. Es werden analytische Ausdrücke der CRU für jeden geophysikalisch relevanten Parameter hergeleitet. Diese Ausdrücke ermöglichen es uns, die Quellen der Unsicherheit zu verstehen, welche die Schätzgenauigkeit der Wellenfeldparameter begrenzen. Es werden stati- stische Modelle für Love- und Rayleigh-Wellen berücksichtigt, welche auf Translationsmessungen, Rotationsmessungen und gemeinsamen Messung der 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, wie die Geschwindigkeit und Richtung der Ausbreitung, sowohl bei niedrigem und hohem SRV beeinﬂusst. Eine Kostenfunktion wird hergeleitet, welche für die Optimierung der Arraygeometrie geeignet ist, dies mit einem besonderen Schwerpunkt auf die Schätzung der Wellenzahlen von sowohl Love wie auch Rayleighwellen. Mehrere computergestützte Lösun- gen zur Minimierung der Kostenfunktion werden vorgestellt und mitein- ander verglichen. Numerische Experimente bestätigen die Wirksamkeit der vorgeschlagenen Kostenfunktion und die daraus resultierende Form der Arraygeometrie, was zu stark verbesserten Schätzungen verglichen mit beliebigen Arraygeometrien sowohl bei niedrigem und hohem SRV führt. Contents

Abstract ii

Kurzfassung iv

Contents vi

1 Introduction 1 1.1 RelatedWorkandBackground ...... 2 1.2 Motivation ...... 7 1.3 Contributions...... 7 1.4 Outline ...... 8

2 Array Processing for Seismic Surface Waves Using Factor Graphs 10 2.1 Introduction...... 10 2.2 SeismicWaveﬁeld...... 12 2.3 AnalysisofSurfaceWaves ...... 16 2.4 Modelling Surface Waves with Factor Graphs ...... 20 2.5 NumericalExamples ...... 24 2.6 ConclusionsandOutlook ...... 27

3 Seismic Waves Estimation and Waveﬁeld Decomposition 40 3.1 Introduction...... 41 3.2 SystemModelandProblemStatement ...... 42 3.3 ProposedTechnique ...... 45 3.4 NumericalResults ...... 49 3.5 Conclusions ...... 59 3.6 Acknowledgments...... 61

4 Processing of Translational and Rotational Motions of Sur- face Waves 63 4.1 Introduction...... 64 4.2 SystemModel...... 66 4.3 TheoreticalPerformanceAnalysis...... 72 4.4 ProcessingTechnique...... 80 4.5 NumericalResults ...... 81 4.6 Conclusions ...... 92 4.7 Acknowledgments...... 93

vi Contents vii

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

5 Sensor Placement for the Analysis of Seismic Surface Waves 96 5.1 Introduction...... 96 5.2 SystemModel...... 99 5.3 SourcesofError ...... 101 5.4 ProblemStatementandDesignCriterion ...... 107 5.5 ArrayDesignMethods ...... 110 5.6 NumericalResults ...... 113 5.7 Conclusions ...... 120 5.8 Acknowledgments...... 121 5.A Relationship Between Likelihood Function and Sampling Pattern 121 5.B SomeRemarksontheMOIsofanArray ...... 124 5.C MIPArrayLayouts...... 125

6 Conclusions 129 6.1 A Method for the Analysis of Surface Waves ...... 129 6.2 Sensor Placement for the Analysis of Seismic Surface Waves . . 132 6.3 Outlook ...... 133

A Seismic Waves Estimation and Waveﬁeld Decomposition with Factor Graphs 135 A.1 IntroductionandSystemModel...... 135 A.2 SeismicWaveﬁeld...... 136 A.3 ProposedTechnique ...... 138 A.4 NumericalExamples ...... 140 A.5 Conclusions ...... 141

B Multi-Sensor Estimation and Detection of Phase-Locked Si- nusoids 143 B.1 Introduction...... 143 B.2 Computing Likelihoods with Factor Graphs ...... 145 B.3 Connection with the Discrete Fourier Transform ...... 148 B.4 NoiseVarianceEstimation...... 148 B.5 ExtensiontoWaveSuperposition ...... 149 B.6 Conclusion ...... 150

C Estimation of Waveﬁeld Parameters of a Single P Wave at the Free Surface 151 C.1 Introduction...... 151 C.2 SystemModel...... 151 C.3 ParameterEstimation ...... 153 C.4 NumericalResults ...... 153 C.5 Discussion...... 158

D Sensor Placement for the Analysis of Seismic Surface Waves - Supplement 159 D.1 Overview ...... 159 DetailsofOptimisedArrays ...... 205 Contents viii

E Derivation of the Cramér-Rao Bounds for Love and Rayleigh Waves 205 E.1 Overview ...... 205 E.2 Derivation of Fisher Information Matrices ...... 208 E.3 DerivationofCramér-RaoBounds ...... 217

Bibliography 222

Acknowledgements 230

About the Author 231 ShortBiography ...... 231 Chapter 1

Introduction

The analysis of seismic surface waves plays a major role in the understanding of geological and geophysical features of the subsoil. Indeed seismic wave attributes such as velocity of propagation or wave polarization reflect the properties of the materials in which the wave is propagating. Seismic surveying methods 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 surface waves and body waves. Love waves and Rayleigh waves are surface waves which exhibit different polarization and, in general, propagate with different velocities. Both velocity of propagation and certain polarization attributes typically change with frequency. The relationship between phase velocity of a wave and frequency is called dispersion relation or dispersion curve (Aki & Richards, 1980). The knowledge of these wave properties allow geophysicists to gain insight into the structure of the subsoil avoiding more expensive invasive techniques (e.g., borehole techniques). A myriad of applications benefit from the knowledge about the subsoil gained through seismic surveys. Seismic hazard studies are an important application of the analysis of surface waves with direct impact on damage mitigation and earthquake preparedness. Many different techniques have been developed for the analysis of surface waves. They can be classified according to several criteria including: the nature of the source of the waves employed; whether relying on a single sensor or on multiple sensors; type of sensors used, e.g. geophone, triaxial seismometers, or rotational sensors; assumptions on specific properties of the wavefield. Each method presents distinctive advantages and is more suited to a specific application than others. In this thesis, we are interested in the development of signal processing methods for the analysis of surface waves. A part of the present work is devoted to the analysis of ambient vibrations using array of sensors. Another part of this thesis deals with the joint processing of translational motion and rotational motion 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 also addressed in this thesis.

1 2

1.1 Related Work and Background

In this section, selected techniques for the analysis of surface waves are introduced. Later in this section, concepts from statistical signal processing and estimation theory that are used in this thesis are presented. One important application of analysis of surface waves is geotechnical site characterization (Tokimatsu, 1997), used in engineering seismology. The goal of site characterization is to determine characteristics of the ground near the surface of the earth in order to estimate amplification of an incident seismic wavefield. Of particular interest are amplification of shear waves and the properties of seismic 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 that a site of soft sediments (exhibiting low seismic velocities) can greatly amplify the wavefield generated by an earthquake compared to a rock site (having high velocities). Therefore the knowledge of soil properties is of great importance in the design of earthquake resistant buildings and in the assessment of existing infrastructure. To the aim of site characterization, seismic waves travelling near the surface of the earth, surface waves, can be studied. The ground motion induced by seismic waves is recorded by seismic instrumentation. From such recordings, the frequency dependence of the wave velocity of surface waves, the dispersion relation, is estimated using signal processing techniques. In a last step, a structural earth model matching the estimated properties of the wavefield is inferred. More precisely, a model of the seismic wave velocities within the upper layers of the earth, compatible with the observed dispersion relation, is found 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 surface waves. Each approach has intrinsic features that make it suited for specific applications. A comprehensive review and classification of surface waves analysis can be found in Rost & Thomas (2002) and Wapenaar et al. (2006). We limit ourselves to reviewing techniques related to this work.

Seismic Source One way to diﬀerentiate techniques used in the analysis of seismic surface waves concerns the source of the waves analysed. Certain techniques employs an active, controlled, source of energy such as a sledgehammer or an explosion. Other techniques exploit seismic waves generated by an earthquake. A latter group of techniques use ambient vibrations, very low amplitude seismic waves generated 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 single station method is the H/V ratio technique which has been widely used for diﬀerent purposes (Fäh et al., 2003; Bonnefoy-Claudet et al., 2006a). The 3

H/V ratio is used as a proxy for Rayleigh wave ellipticity and to determine the SH resonance frequency. Methods that attempt to estimate Rayleigh wave ellipticity 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 use of array of seismic sensors and array processing techniques (Van Trees, 2002). The use of array processing techniques in seismology has a long history. Early applications of array processing techniques to seismology date back several decades (Aki, 1957; Capon, 1969; Lacoss et al., 1969). More recent developments allow us to separate Love waves and Rayleigh waves (Fäh et al., 2008). The most important assumption necessary for all these techniques is the presence of plane wave fronts, which implies an underlying layered earth model (1D model). Seismic interferometry methods, are another class of techniques that rely on the cross-correlations between pairs of sensors (Wapenaar et al., 2006). These methods exploit the relationship existing among the cross-correlations between a 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 have been applied in different settings including the analysis of coda waves (Campillo & Paul, 2003) and of ambient vibrations (Shapiro et al., 2005). Typically, important requirements for this class of techniques are the diffuse nature of the wavefield and an even distribution of the sources at every azimuth. Both array processing techniques and interferometry techniques lead to the retrieval of the dispersion relation of surface waves. In general, the first class of techniques usually leads to the retrieval of a more accurate dispersion curve in shorter surveying times. Polarization properties of the wavefield may also be retrieved. However the applicability of these techniques is limited to sites where the 1D model assumption holds. In contrast, interferometric techniques are not constrained by such limiting assumption. However they typically require the stacking of a significant number of cross-correlations and thus the availability of very long recordings. By means of a final tomographic reconstruction step, interferometric techniques allow the determination of velocity maps rather than mere 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 of instrumental observations. This is due to both the technical challenges involved in measuring rotational motions and to the widespread belief that rotational motions are insignificant (Gutenberg, 1927; Richter, 1958). In the last years the attention of the seismological community towards the study of rotational motion increased significantly (Lee et al., 2009; Igel et al., 2012). One reason being the availability of direct measurements of rotational motions. The most striking feature of rotational motions is that, together with translational motions, they enable us to estimate velocity of propagation of seismic waves from a point measurement. The amount of rotational motion induced by 4 a seismic wave is inversely proportional to the wavelength, and is thus related to the velocity of propagation. As a result, a six-components measurement of both translational and rotational motions at a single spatial location gather sufficient 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 single sensor location has been also addressed and it has been shown that the retrieval of 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 substantially aﬀected by the physical arrangement of the sensors and their number. In array processing techniques, a poor array geometry may, for example, lead to a large uncertainty in the retrieved dispersion curves. The limitations of diﬀerent array geometries have been investigated by different authors, e.g., Woods & Lintz (1973); Asten & Henstridge (1984); Toki- matsu (1997); Kind et al. (2005); Wathelet et al. (2008). In particular, the interest has been to identify a range of wavenumbers, or a related quantity such as velocity or slowness, where the result of the array processing is more reliable. The largest and the smallest resolvable wavenumbers have been related either to the array aperture and the smallest interstation distance or to the height of the sidelobes of the array response function. Qualitative guidelines, based on empirical evidences, for array design are provided in Rost & Thomas (2002) and in Kind et al. (2005).

Statistical Signal Processing In the last decades the availability of seismic recordings increased in terms of quality, quantity, and diversity. Within a similar time frame, the steady improvements in digital electronics provide us with unprecedented computing capabilities. These factors both create the need and enable the development of sophisticated data processing algorithms able to model complex systems exploiting all the available measurements. Statistical signal processing provides eﬀective means to deal with the estimation of the values of parameters from measurements corrupted by random noise. This is a well developed subject, lying at the meeting point between statistics and signal processing. Hereafter we introduce selected aspects of statistical signal processing that relevant to this work and refer the interested reader to Kay (1993) for a more detailed reading.

Parameter Estimation The estimation of parameters of interest relying on noisy observations is a central task in statistical signal processing. An estimator is a procedure that relies on the noisy measurements to provide an estimate of the parameter of interest. Mathematically, an estimator is a function g( ) of the measurements · y˜ providing an estimate θˆ of the value of the unknown parameter θ, i.e., θˆ = g(y˜). The choices about how to design an estimator and the analysis of its performance are the main topics of estimation theory. 5

Different design choices of an estimator are possible and they may lead to different performance. We observe that an estimator is a random quantity, indeed for different noisy observations of y˜ different estimates θˆ are obtained. Therefore, rather than analysing the quality of a single estimate it is more meaningful to evaluate performance of an estimator in an average sense. At least two properties of an estimator are desirable: unbiasedness and small mean-squared estimation error (MSEE). Bias is a measures of the average deviation of the estimate from the true value and is defined as

Bias(θˆ) = E θˆ 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(θˆ) = 0. Otherwise the estimator is said to be biased. The MSEE measures the average of the square of errors and is deﬁned as

MSEE(θˆ) = E (θˆ E θˆ )2 . (1.2) { − { } } It is desirable for an estimator to have small MSEE as this implies smaller fluctuations around the expected value E θˆ . As it will be clarified, the MSEE cannot be made arbitrarily small. { } The Cramér-Rao bound (CRB) is a lower bound on the MSEE of an estimator. For a given statistical model, a fundamental limit exists and the MSEE cannot be made smaller than this bound. Such limit depends on the statistical model and is independent of the estimation technique used and algorithmic implementation. The CRB is therefore a useful benchmark to validate the quality of an estimation algorithm. An analytic expression of the CRB also allows us to understand how the different parameters of the system affect estimation accuracy.

Maximum Likelihood Estimation The maximum likelihood (ML) estimation method is a widely-used and well known estimation technique (Fisher, 1922). Its popularity is due to the fact that it is almost always possible to implement, at least numerically, an ML estimator. Notably, a ML estimator exhibits optimal performance under many settings. In particular, for a sufficient sample size, the ML estimator is unbiased and the MSEE achieves the smallest possible value indicated by the CRB. The main steps in ML estimation are the following. First, it is necessary to formalize mathematically the relationship between the measurements, the parameters of interest, and potentially other unobserved quantities. This step accounts for establishing a statistical model for the measurements and, in particular, defining the probability density function (PDF) of the measurements. We denote with pY (y, θ) the PDF of the measurements where Y are random variables and θ is a deterministic parameter. The second step is to compute the likelihood of the observations. The likelihood function (LF) is readily obtained from the PDF. The LF of the observations is a function of the parameter θ and is defined as

ℓ(θ) = pY (y˜, θ) , (1.3) 6 where y˜ denotes the observed measurements. The ﬁnal step, is to maximize the LF over the parameter space.

θˆML = argmax ℓ(θ) , (1.4) θ where θˆML is the ML estimate of the parameter. The maximization in (1.4) may be tackled by diﬀerent means. At best, an analytic solution may be found. In other cases, an exhaustive search, over a possibly multidimensional parameter space, may be the only available option.

Factor Graphs A complex system where a large number of random variables and statistical parameters interact with complex relationships can be effectively represented by a graphical model (Jordan, 2004). Within the graphical model, observed random variables (measurements), unobserved random variables, and parameters of the statistical model are represented in a unique framework together with the functional relationships occurring among them. In our approach we rely on factor graphs, one flavour among many graphical modelling techniques (Kschischang et al., 2001; Loeliger, 2004; Loeliger et al., 2007). The factor graph can be used to perform inference tasks in an efficient manner. As an example, computing the likelihood of the observations and subsequently ML estimation can be performed exploiting the structure of the graph. Moreover, by inspecting the graph structure it is possible to understand the relationship between the different parts of the stochastic system and then, for example, derive sufficient statistics which enable to efficiently compute statistical quantities of interest. To summarize, the factor graph enables us to define the statistical model pY (y, θ) and to derive efficient algorithms to address inference tasks. Both tasks are of crucial importance in practice and may be not trivial for complex systems.

Spatial Sampling The sampling of spatial signals exhibits some difficulties that are not present in the more common setting of sampling signals in time. One important difference is that when sampling spatially it is not possible to use anti-aliasing filters. Another peculiarity of spatial sampling is that, in certain applications, the number of sensors available to sample the wavefield is very limited and uniform sampling schemes cannot be used. When sampling temporal signals, an analogue anti-aliasing filter is typically used before sampling in order to limit the bandwidth of the signal and prevent aliasing in the sampled signal. Such filtered signal, can then be sampled at a sufficiently high rate according to the Nyquist-Shannon sampling theorem. This approach is however infeasible for spatial signals since the anti-aliasing filter cannot be implemented. When a large number of sensors is available uniform sampling schemes may be used. For example, if the spatial frequency content of the signal is limited to a known disk shaped region, hexagonal sampling is optimal in the sense that the density of the samples is minimized (Vaidyanathan, 1993). However, in a 7 setting where the number of available sensors is very limited a non-uniform sampling scheme may be necessary (Holm et al., 2001). The choice of the location of the sensors is a non-trivial task.

1.2 Motivation

This work aims at improving signal processing techniques for the analysis of surface waves in different directions. In particular, the main goal is to deliver accurate estimates of the geophysical parameters of interest by optimally exploiting the available measurements. The availability of improved estimates of geophysical quantities of interest will provide better constraints for the geophysical 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 tool that can be applied to analyses ambient vibration wavefields in areas of hydro- carbon reservoirs. This includes an optimization of existing methods for array signal processing, and the development of a real-time system for on-site processing. Specific objectives addressed in this thesis include: the identification of different wave types from ambient vibration measurements, the identification of small amplitude waves, and the optimization of array geometry. Array processing techniques for the analysis of seismic surface waves currently in use presents some limitations and therefore have potential for improvements. In several recent applications of array processing to the analysis of ambient vibrations, only the vertical component of the seismometer has been used (Cornou et al., 2003a; Wathelet et al., 2008). In other work, the horizontal and the vertical component are processed separately and Love waves are distinguished from Rayleigh waves (Fäh et al., 2008). In Poggi & Fäh (2010) Rayleigh wave ellipticity is estimated, however without modeling the retrograde or prograde particle motion. A rigorous joint treatment of all measurements and all wavefield parameters is lacking in literature, and has been addressed in this thesis. The analysis of surface waves employing measurements of rotational motions is presently limited to the analysis of Love waves from single sensors (Igel et al., 2005; Ferreira & Igel, 2009). In these works it is shown that the retrieval of Love wave dispersion from a single sensor is possible. To the best of our knowledge, there are no techniques for the analysis of Rayleigh waves from single sensor and there are no techniques to jointly process rotational and translational motions from array recordings. This is another main point addressed in the thesis. The design of array geometries has not been addressed quantitatively by the ambient vibrations community. It is necessary to develop quantitative criterion and algorithms for sensor placement and move beyond the current qualitative guidelines 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 analysis 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 parameters of Love waves and Rayleigh waves together with all the measurements are jointly modelled within the proposed framework. The method ensures an optimal usage of the available measurements according to the ML criterion. In particular, besides the estimation of Love wave and Rayleigh wave phase velocity, the proposed method accounts for the retrieval of Rayleigh wave ellipticity together with the retrograde or prograde sense of rotation of the wave. The method also deals with the simultaneous presence of multiple waves, possibly of different type. This enables to decompose the wavefield and to detect weaker waves otherwise covered by stronger waves. The proposed method accounts for sensors with different noise level by merging measurements of different quality properly. This is critical when using together sensors of different kind such as translational sensors and rotational sensors. Performance of the proposed method are first evaluated on synthetic datasets by means of Monte Carlo simulations. It is shown that the actual implementation exhibits the smallest achievable MSEE indicated by the CRB for a sufficiently large signal-to-noise ratio (SNR). Performance are also assessed on field measurements of ambient vibrations from sensor arrays. The joint processing of translational and rotational motions is tested on recording from controlled explosions. The case of a single six components sensor and the case of an array of three and six components sensors are considered. The proposed method is flexible enough to be able to model different wave types with minor modifications. For example, it is shown on synthetic data how it is, in ideal conditions, possible to retrieve both P and S wave velocity from measurements of a single incident P wave. The impact of array geometry on the proposed method is also studied. The different sources of error related to the geometry of the sensor array are reviewed in detail for the specific cases of Love waves and Rayleigh waves. A quantitative design criterion is proposed together with sensor placement algorithms. Usefulness of the design criterion and effectiveness of the considered sensor placement algorithms are validated by means of Monte Carlo simulations. Other contributions of this thesis includes the derivation of analytical expressions the CRBs of the quantities of geophysical interest. This provides a benchmark for the evaluation of algorithms and understanding on how the different system parameters affect estimation accuracy.

1.4 Outline

This thesis is composed of a collection of independent manuscripts and is organized as follows. In Chapter 2, a tutorial style article introduces aspects of the analysis of surface waves and their modelling using factor graphs. Chapter 3 presents a ML approach for the analysis of Love waves and Rayleigh waves using translational triaxial seismometers. Applications to synthetic datasets and ﬁeld measurements of ambient vibrations are shown. Addi- tional details concerning the design of the factor graph underlying the proposed method are given in Appendix A and Appendix B. Appendix C describes the modelling of a single incident P wave using the proposed method. 9

In Chapter 4, the ML method is extended to address the joint processing of translational and rotational motions. Applications to the analysis of seismic surface waves and the joint processing of rotational and translational measurements are shown. In Chapter 5, the impact of array geometry on the estimation system is reviewed, sensor placement criterion are proposed, and an algorithm for sensor placement is presented and evaluated. In Appendix D, a list of arrays obtained using the proposed sensor placement method for diﬀerent numbers of sensors is given. Appendix E includes additional details of the derivations of Fisher information matrices for the statistical models considered in the thesis and the Cramér-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-Andrea Loeliger2

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 information about geological and geophysical features of the subsoil. Depending on the application, surveying methods consist of different measurement setups and involve different signal processing techniques. In this work, we are interested in the analysis of seismic surface waves for geotechnical site characterization. In particular, we focus on the processing of array recordings from ambient vibrations. After reviewing array processing techniques of current use for the processing of surface waves, we give an introduction to factor graphs, tailored to the analysis of wavefields. The factor graph approach provides a framework for maximum likelihood parameter estimation of any wave type in an efficient manner. In addition, in the same framework, we explain how to address wave superposition by gradually decomposing the wavefield. We show numerical examples of the processing of ambient vibrations recordings from arrays of seismic sensors.

2.1 Introduction

Analysis of the seismic waveﬁeld enables us to gather knowledge of geological and geophysical features of the subsoil. Indeed, seismic wave attributes such as velocity of propagation or polarization reﬂect the properties of the materials in 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 invasive techniques (e.g., borehole techniques) and therefore represent a valuable tool in 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 characterization (Tokimatsu, 1997), used in engineering seismology. The goal of site characterization is to determine characteristics of the ground near the surface of the earth in order to estimate amplification of an incident seismic wave field. Of particular interest are amplification of shear waves and the properties of seismic 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 that a site of soft sediments (low velocities) can greatly amplify the wavefield of an earthquake compared to a rock site (high velocities). Therefore the knowledge of soil properties is of great importance in the design of earthquake resistant buildings and in the assessment of existing infrastructure. To the aim of site characterization, seismic waves traveling near the surface of the earth, surface waves, can be studied. The ground motion induced by seismic waves is recorded by seismic instrumentation. From such recordings, the frequency dependence of the wave velocity of surface waves, the dispersion relation, is estimated using signal processing techniques. In a last step, a structural earth model matching the estimated properties of the wavefield is inferred. More precisely, a model of the seismic wave velocities within the upper layers of the earth, compatible with the observed dispersion relation, is found by solving a non linear geophysical inverse problem (Tarantola, 2004). Seismic surveying methods for site characterization can be classified into active and passive methods. Active methods require a controllable source of energy such as an explosion or a sledgehammer. Common active surface wave methods 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 by natural sources such as ocean waves, wind or atmospheric changes and by an- thropogenic sources (Bonnefoy-Claudet et al., 2006b). The advantage of passive methods is their applicability to urban areas and the ability to analyze lower frequencies that are very difficult to excite with active techniques, thus allowing to resolve deeper structures in the earth (Okada, 1997). The purpose of this paper is two-fold. First, we introduce the reader to the seismic wavefield and to array processing techniques in use for the analysis of surface waves. Our interest is on passive surveying method for site characterization. Nonetheless, parts of the material presented are valid in other settings. We focus on array processing techniques and we do not explain the details of the aforementioned geophysical inverse problem. Second, we provide an introduction to factor graphs with emphasis on the techniques necessary for the analysis of wavefields. We show how the framework provided by factor graphs allows the statistical modeling of complex systems in an effective manner. This paper is organized as follows. We give an introduction to the seismic wave field of surface waves and the sensors in use to measure it. We review the methodology in use for the analysis of surface waves, in particular array processing methods used for the analysis of ambient vibrations. The second part 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 κ

(a) A Rayleigh wave with retrograde (b) A Love wave. particle motion.

Figure 2.1: A depiction of the displacement induced by surface waves. Both waves travel in the κ direction. of wavefield parameters in an efficient way. We provide an introduction to factor graphs, with a focus on the tools necessary for the study of wavefields, together with an application to surface waves. Finally, we provide numerical examples of 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 ( )∗ for ∗ complex conjugation, ( )T for transposition, and ( )H = ( )T for Hermitian· transposition. We loosely· stick to the convention of· using upper-case· letters for random variables (Y , X) and lower-case letters for a realization of a random variable (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 of a seismic wave will cause the particle to oscillate and rotate around its resting position. In an elastic medium, the particle will return to its original position after the passage of the wave. In general, the motion of a particle is completely described by six quantities, namely the displacements along three orthogonal axes and the rotations around these axes. An important characteristic of the seismic wavefield is the simultaneous presence of several waves of possibly different type. An important distinction among wave types is between body waves and surface waves. Waves propagating in the interior of the earth are called body waves and are further classified as primary waves (P waves) or secondary waves (S waves). P waves are compressional waves like the acoustic waves: the particle motion induced 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 direction of propagation. In the elastic case, body waves are non-dispersive, i.e., they propagate at each frequency with the same velocity. Anelasticity also introduces body wave dispersion. Surface waves propagate on the surface of the earth, and their energy is concentrated near the surface. The lower the frequency of a surface wave the deeper is the penetration depth into the earth. Among surface waves we distinguish between Rayleigh waves and Love waves. As explained in more 13

0 P wave velocity -5 S wave velocity

-10 ] m [ -15

Depth -20

-25

-30 0 500 1000 1500 2000 Velocity [m/s] (a) P wave and S wave velocities for a structural model of a single layer over an inﬁnite half-space.

1000 Fundamental mode First higher mode 800 Second higher mode s] /

[m 600 κ ω

400 Velocity 200

0 0 2 4 6 8 10 12 Frequency [Hz] (b) Dispersion curve for Rayleigh waves.

1000 Fundamental mode First higher mode 800 Second higher mode s] /

[m 600 κ ω

400 Velocity 200

0 0 2 4 6 8 10 12 Frequency [Hz] (c) Dispersion curve for Love waves.

Figure 2.2: A simple layered-earth model 2.2(a) and the corresponding dispersion 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 propagation depends on the frequency. The retrieval of the dispersion relation (i.e., how velocity depends on frequency) is often the main goal of seismic surveying methods using surface waves. Rayleigh waves and Love waves propagate in general with different velocities. In addition, they propagate in modes, i.e., at a given frequency the wave can propagate only with a discrete number of specific velocities. Fig. 2.2(a) shows the P wave and S wave velocities for a structural model consisting of a 25 meters thick layer of soft sediments with low seismic velocities and an underlying bedrock with higher velocities. Body wave velocities together with the density of the materials (not shown) suffice to characterize a simple layer over half-space model. Such a model is representative of what is typically found in river basins, alpine valleys or lake shores. Remember that the motivation to studying surface waves is that their properties, such as dispersion curve and polarization, carry information about the structure of the subsoil. 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 quantities such as translations and rotations of the ground particle. Most widely used in the seismological community are triaxial velocimeters. A single triaxial velocimeter is capable of measuring the ground velocity along the three orthogonal directions. Therefore such sensors are sensitive to translation and not to rotation. 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 ground rotations induced by the passage of a seismic wave (Lin et al., 2009). In this paper, we restrict ourselves to the translational motions induced by seismic surface waves. To measure surface waves, an array of triaxial velocimeters is deployed on the surface of the earth. We restrict our interest to small aperture 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 the coordinate 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- ∂u surement is the velocity field ∂t .The displacement of the ground at position p R3 and time t can be described by the vector field ∈ u(p, t) , u (p, t), u (p, t), u (p, t) : R4 R3 . (2.1) x y z → 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 the earth (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 (cos ψ, sin ψ, 0)T, whose magnitude κ is the wavenumber. The velocity of propagation of the wave is ω/κ. We denote the amplitude and phase of the wave by α and ϕ, respectively. 15

ψ y x p

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

Rayleigh waves exhibit an elliptical particle motion conﬁned to the vertical plane perpendicular to the surface of the earth and containing the direction of propagation of the wave. The particle displacement generated by a single Rayleigh wave is

u (p, t) = α sin ξ cos ψ cos(ωt κTp + ϕ) x − u (p, t) = α sin ξ sin ψ cos(ωt κTp + ϕ) (2.2) y − u (p, t) = α cos ξ cos(ωt κTp + π/2 + ϕ) . z − The ellipticity angle ξ determines the eccentricity of the elliptical motion and the 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 parameterize ±Rayleigh waves with the parameter vector θ , (α, ϕ, κ, ψ, ξ). Fig. 2.1(a) depicts the displacement induced by a Rayleigh wave. Love waves exhibit a particle motion conﬁned to the horizontal plane, the particle oscillates perpendicular with respect to the direction of propagation. The particle displacement generated by a single Love wave is

u (p, t) = α sin ψ cos(ωt κTp + ϕ) x − − u (p, t) = α cos ψ cos(ωt κTp + ϕ) (2.3) y − 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 to be composed of the linear superposition of several Rayleigh and Love waves. Only certain elements in the parameter vector θ are of interest to the geophysicist. The quantities α, ϕ, and ψ depend on the source while the quantities κ 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 of source-independent quantities which are directly related to the geology of the specific site. In other applications, such as the seismic verification of nuclear explosions, 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 processing techniques for the estimation of physical quantities of interest such as velocity (or alternatively wavenumber) and polarization attributes. The processing is repeated separately at different frequencies and in different time windows. The next step is to retrieve the dependence of these quantities on the frequency. Afterwards, these dependencies can be used in the last step, called geophysical inversion, to infer a structural model.

Overview of Array Processing Techniques For the analysis of surface waves, a planar sensor array is typically deployed and array processing techniques are used (Wathelet et al., 2008; Di Giulio et al., 2006). Array processing techniques used in geophysics are similar to the techniques 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 different setting is referred to as frequency-wavenumber analysis. In fact, velocity is often the most relevant parameter for the geophysicist. Other distinctive traits of the seismic wavefield are the wave polarization and the simultaneous presence of several waves of different type. Early applications of array processing techniques 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 the wavefield does not change over the observation time. In addition, we restrict the analysis to a specific frequency ω. Consider a source emitting a monochromatic signal α cos(ωt + ϕ). The sensor array observes L distinct signals at discrete times tk for k = 1,...,K as

i(ωtk+ϕ) Y k = Re a α e + Zk , (2.4) , (1) (L) T where Y k Yk ,...,Yk . Measurements are corrupted by zero mean 2 2 additive Gaussian noise Zk with diagonal covariance matrix diag(σ1 , . . . , σL). The vector a CL is called the steering vector and models how the source signal is scaled∈ and delayed at each sensor. E.g., let the ℓ-th element of the iφℓ (ℓ) (ℓ) steering vector be ρℓ e , then Yk = α ρℓ cos(ωtk + ϕ + φℓ) + Zk . In general, the steering vector depends both on properties of the waveﬁeld (e.g., wave velocity, polarization) and on characteristics of the array (e.g., position of the sensors, properties of the sensors). The beamforming techniques usually employed in geophysics operate in the frequency domain. From the ℓ-th signal the frequency snapshot Y˘ℓ is computed using the discrete-time Fourier transform as

K ˘ , (ℓ) −iωtk Yℓ Yk e . (2.5) Xk=1 The spectral spatial covariance matrix is deﬁned as

H R , E Y˘ Y˘ , (2.6) h i 17

˘ T ˆ where Y = Y˘1,. .., Y˘L . An estimate R of the spectral spatial covariance matrix can be obtained with the sample covariance matrix having elements W 1 [Rˆ ] , y˘(w)(y˘(w))∗ , (2.7) j,ℓ W j ℓ w=1 X where w indexes different observations y˘j taken at different time windows (Van Trees, 2002). Now, we recall the DOA estimation problem in the simple case of a uniform linear array (ULA). Consider a plane wave traveling with known velocity ω/κ through a ULA of N scalar sensors (i.e. L = N) with inter-sensor distance d. The array steering vector a is defined as

T a(ψ) , 1, e−iκd cos ψ, . . . , e−i(N−1)κd cos ψ . (2.8)

The complex argument of each element of the steering vector corresponds to the phase delay with respect to the ﬁrst element for an incoming wave with azimuth ψ. In order to estimate the azimuth, the Bartlett beamformer method matches the phase delay experienced by the incoming wave at diﬀerent sensors and then sum up the delayed signals to form the beamformer output (Krim & Viberg, 1996). The Bartlett beamformer is

a(ψ)H Rˆ a(ψ) P (ψ) , . (2.9) a(ψ)H a(ψ)

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 vector corresponding to the true ψ the frequency snapshots Y˘ℓ ℓ=1,...,L are summed in-phase in (2.9) and the beamformer output is thus maximized.{ } This method is also referred to as classical beamforming. In the seismic case, often we do not restrict ourselves to ULAs, but use arbitrary planar arrays, and the velocity of propagation is not known. For an array of N sensors at positions pn n=1,...,N the steering vector is deﬁned as a function of both azimuth ψ and{ wavenumber} κ as

κTp κTp T a(ψ, κ) , e−i 1 , . . . , e−i N . (2.10)

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

a(ψ, κ)H Rˆ a(ψ, κ) P (ψ, κ) , . (2.11) a(ψ, κ)H a(ψ, κ)

Again, the estimates of ψ and κ are found by maximizing P (ψ, κ). Note that in both the acoustic setting and in the electromagnetic setting, the velocity of propagation of the wave is known and the maximization of P (ψ, κ) is performed only on the azimuth. Other array processing techniques widely known in the radar and antenna communities such as the Capon method (Capon, 1969) and the MUSIC algorithm (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 interfering signals. The output of the Capon beamformer is 1 P (ψ, κ) , . (2.12) a(ψ, κ)H Rˆ −1 a(ψ, κ) The main advantage of the Capon beamformer is its ability to resolve closely spaced sources which may be seen as a single source by the Bartlett beamformer (Krim & Viberg, 1996). The multiple signal classification (MUSIC) algorithm is also able to distinguish closely related sources. In the MUSIC algorithm the spectral spatial covariance matrix R is decomposed via eigenvalue decomposition. Let λˆ and vˆ be the eigenvalues, sorted in decreasing order, { ℓ}ℓ=1,...,L { ℓ}ℓ=1,...,L and the eigenvectors of Rˆ , respectively. In presence of M signals, the first M eigenvalues are called signal-subspace eigenvalues and the remaining L M are called noise-subspace eigenvalues. It is observed that the noise eigenvectors− are ⋆ H ⋆ orthogonal to the true steering vector a , i.e., vℓ a = 0. The estimates for ψ, κ are found maximizing the MUSIC pseudo spectrum 1 P (ψ, κ) , , (2.13) a(ψ, κ)H Πˆ a(ψ, κ) Πˆ , L ˆ ˆH where ℓ=M+1 vℓ vℓ . Moreover, a method for estimating the number of waves M is required. In (Cornou et al., 2003a), the MUSIC algorithm is used for the analysisP of ambient vibrations using the vertical component. In a more general setting, we are interested not only in azimuth and wavenumber but also in further properties such as wave polarization. Moreover, we also want 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 number of sensors N, has been referred to as vector-sensor beamforming (Hawkes & Nehorai, 1998). For a generalized beamformer, we define the steering vector a(θ), accounting for arbitrary scalings and delays, as a function of the wavefield parameters θ. This beamformer output depends on the wavefield parameters and can be written as a(θ)H Rˆ a(θ) P (θ) , . (2.14) a(θ)H a(θ) As a concrete example, for a Love wave the steering vector a C2N has the form ∈ T κTp κTp a(ψ, κ) , sin ψ e−i 1 , cos ψ e−i 1 ,... . (2.15) − Eq. (2.3) supports this definition of steering vector. The steering vector for a Rayleigh 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 used also in this∈ setting. In recent work (Donno et al., 2008), a vector-sensor technique for the analysis of seismic waves using jointly the three components is proposed. First, polarization and velocity parameters are estimated, then amplitude is estimated by linear regression. In (Fäh et al., 2008), a technique able to analyze also the horizontal components was proposed. The technique is applied to seismic surface waves to 19 distinguishing between Rayleigh and Love waves. Vertical and horizontal components 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 spatial autocorrelation method (SPAC) (Aki, 1957; Köhler et al., 2007). In SPAC, properties of the cross-correlation between the signals recorded at pairs of sensors are exploited. This method requires a large number of seismic sources uniformly distributed around the array. Later in this paper we focus on a method for ML estimation of wavefield parameters and wavefield decomposition that was proposed by the authors in (Maranò et al., 2011). It can be shown that the general Bartlett beamformer of (2.14) is proportional to the likelihood function when the temporal sampling is uniform and the noise variances σ2 are equal. { ℓ }ℓ=1,...,L Retrieval of Dispersion and Ellipticity Curves We have shown that using array processing techniques, physical quantities of interest such as the wavenumber κ or ellipticity angle ξ can be estimated. The next goal is the retrieval of the dispersion curves (for both Rayleigh and Love waves) and of the ellipticity curve (only for Rayleigh waves). Such curves describe how the physical quantity of interest changes with frequency. The estimates of the wavenumber and ellipticity angle obtained at each frequency and in each time window using array processing techniques must be combined. Seismic wavefields found in nature have spectral characteristic depending on the seismic source. Different sources are able to excite different wave types in different ranges of frequencies (Bonnefoy-Claudet et al., 2006b). The quantities of interest are often unknown functions of frequency (e.g., velocity or polarization as a function of frequency) and different frequencies are typically processed independently. As an example, Fig. 2.4 shows the outcome of processing the same data (ten seconds) at six different frequencies and a plot summarizing how wavenumber changes with frequency. Note that the dispersion relation can be equivalently represented as in Fig. 2.2(b) (velocity vs. frequency) or as in Fig. 2.4(g) (velocity vs. wavenumber). During a seismic survey, sources vary over time. Commonly, a long-time recording is split into shorter windows. Inside each window the seismic sources are assumed to be constant. Each window is processed independently and the parameters estimated in the different time windows are jointly represented in graphical form at the end of the processing by means of, as an example, an histogram. An expert can individuate the dispersion or ellipticity curves by visual inspection of such depictions.

Geophysical Inversion The last stage of seismic surveying consists to finding a structural model that is able to explain the observed properties of the seismic wavefield (Tarantola, 2004). In the analysis of surface waves, these properties are the observed dispersion curves and ellipticity curves (Wathelet et al., 2008; Wathelet, 2008; Fäh et al., 2003). Often a layered-earth model is used. Such a model is described by the 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 structural model, based on the observed dispersion curves and ellipticity curves, poses a non-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 of the site can help to further constraint the inversion and to select a meaningful model.

2.4 Modelling Surface Waves with Factor Graphs

Introduction We quickly recall that the aim of array processing techniques in the context of this paper is to find ML estimates of a wave parameter vector θ given the measured data. For Rayleigh waves θ = (α, ϕ, κ, ψ, ξ) and for Love waves θ = (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 wave ellipticity. Under certain conditions, the general beamformer output (2.14) is equivalent with a likelihood function for this parameter vector under the statistical model in Eq. (2.4). In principle, this equivalence can be derived using classical tools of statistics (Kay, 1993). In this section we present an alternative approach to ML estimation (Maranò et al., 2011; Reller et al., 2011), in which we use factor graphs and sinusoidal state-space models (see boxes “Factor Graphs and 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 particularly easy to derive. Even more importantly, this approach allows us to highlight underlying principles which may be used in related problems. Last but not least, factor graphs provide a unified framework for many other signal processing tasks (Loeliger et al., 2007) and we believe that this approach will foster further 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 all sensors for a given frequency ω in a single factor graph. (See box “Factor Graphs and Likelihood” for an introduction to the notion of a factor graph.) This factor graph, shown in Fig. 2.5(a), consists of one sinusoidal state-space model per signal (See box “A State-Space Model for Noisy Sinusoids”, cf. Fig. S.3), and one additional factor gθ, the glue factor. We name gθ a glue factor because it connects all the individual state space models at time t0. The only factor depending on the wave parameter vector θ is the glue factor gθ. This fact will enable 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 noisy version of the scaled and phase shifted sinusoid α cos(ωtk + ϕ), where α and ϕ are the amplitude and the phase of the wave. The ℓ-th signal takes the form

(ℓ) (ℓ) Yk = α ρℓ cos(ωtk + ϕ + φℓ) + Zk , (2.16) 21

(ℓ) 2 where Zk is zero-mean white Gaussian noise with variance σℓ . For every wave type of interest we can deﬁne a mapping Γ: θ ρℓ, φℓ ℓ=1,...,L relating the wave parameter vector to the amplitude scalings7→ and { phase} shifts. For a Rayleigh wave as in Eq. (2.2), this mapping is

(ρ , φ ) = sin ξ cos ψ, κTp , 3n−2 3n−2 − n (ρ , φ ) = sin ξ sin ψ, κTp , (2.17) 3n−1 3n−1 − n (ρ , φ ) = cos ξ, κTp + π/2 , 3n 3n − n and for a Love wave as in Eq. (2.3), this mapping is

(ρ , φ ) = sin ψ, κTp , 3n−2 3n−2 − − n (ρ , φ ) = cos ψ, κTp , (2.18) 3n−1 3n−1 − n (ρ3n, φ3n) = (0, 0) , where n = 1,...,N enumerates the sensors. In principle, such a mapping can be deﬁned for arbitrary sinusoidal waves. Note that in any such mapping, α and ϕ do not feature. In the factor graph of Fig. 2.5(a),

(ℓ) cos(ωtk + ϕ + φℓ) Xk = α ρℓ (2.19) sin(ωtk + ϕ + φℓ)! is the state vector of the ℓ-th sinusoidal state space model. For ease of notation , (ℓ) we denote the state vectors at time t0 by Sℓ X0 . Similarly, let

cos ϕ s˜(θ) , α (2.20) sin ϕ! be the state vector of the wave at time t0. With these deﬁnitions we now model the coupling between the signals by means of Dirac delta constraints in the glue factor L gθ(s ,..., s ) = δ s H (θ) s˜(θ) , (2.21) 1 L ℓ − ℓ ℓ=1 Y where the coupling matrices

cos φℓ sin φℓ Hℓ(θ) , ρℓ − (2.22) sin φℓ cos φℓ ! depend on θ via the mapping Γ. The internals of the glue factor (2.21) are depicted 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) of all the | k k=1,...,K,ℓ=1,...,L y (ℓ) state-space models and contains all the measured signals yk k=1,...,K,ℓ=1,...,L. 22

Maximum Likelihood Estimate The goal is to make an ML estimate

θˆ = argmax f(y θ) (2.23) θ | of the parameter vector θ. The factor graph in Fig. 2.5(a) with Figs. S.3 and 2.5(b) inserted allows us to derive the likelihood function f(y θ) in terms of sum-product messages on any edge in this graph as in Eq. (S.4).| We choose the edge S˜ in Fig. 2.5(b) and hence get

→ ˜ ← ˜ ˜ f y θ = −µS˜(s) −µS˜(s) ds (2.24) Z˜ s ← ˜ = −µS˜ s(θ) (2.25) L ← = gθ(s ,..., s ) µ (s ) ds ds , (2.26) ··· 1 L −Sℓ ℓ 1 ··· L Z Z ℓ=1 s1,...,sL Y where the second equality is due to the fact that s˜(θ) is fixed and the third equality 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 observations ← y into messages −µSℓ using sum-product message passing, and then ℓ=1,...,L← use (2.26). In other words, µ is a sufficient statistic for estimating −Sℓ ℓ=1,...,L θ. Note that this statement in principle applies to more general models, as long as their PDF factorizes as in Fig. 2.5(a). ← At first, Eq. (2.26) looks unwieldy, but for Gaussian messages −µSℓ and factors as in Table 2.1, message update tables from (Loeliger et al., 2007) can

Local function Factor graph symbol

X Gaussian PDF x 0, σ2 σ (0, 2) N N X Z = Equality con- δ(x y)δ(x z) straint − − Y

Y Sum constraint δ(z x y) X Z − − +

X Y Linear con- δ(y Ax) A straint −

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 and variance σ2 by m, σ2 . · N ·

← be used. These tables show, how the inverse covariance matrix W−˜ and mean ← ← ← ← S m H θ W m vector −S˜ of −µS˜ can be expressed in terms of ℓ( ), −Sℓ , −Sℓ ℓ=1,...,L. For the maximization of Eq. (2.25) over θ, we note that s˜ depends on α and ϕ only, while H depends on the remaining parameters only. Since Eq. (2.20) formulates a one-to-one mapping between (α, ϕ) and s˜, we can find ML estimates of (α, ϕ) by first finding an ML estimate of s˜ followed by inverting ← the mapping. The ML estimate of s˜ is, however, simply the mean m of the ← −S˜ message µ . −S˜ ← ← The resulting, partially maximized expression −µS˜ m−S˜ can still be written ← ← H θ W m in terms of ℓ( ), −Sℓ , −Sℓ 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 the 2 noise variances σℓ ℓ=1,...,L are equal for all signals. The final maximization over the remaining parameters is non-convex and can be done using a grid search followed by a gradient ascent method.

Noise Variance Estimation and Superposed Waves In this section, we consider two extensions which can have a considerable impact on the performance of the estimation of waves parameter vectors θ. In contrast to the previous section, we now describe iterative algorithms for approximate ML estimation. Despite their sub-optimality, these algorithms yield good results. 2 In general, the noise variances σℓ ℓ=1,...,L can vary considerably among the 2 signals. In our setup, an increase in σ ℓ can model not only measurement noise but also sensor failure, sensor misplacement, interfering signals and weaker 2 superposed waves. In these circumstances, estimating σℓ in most cases leads to an improved weighting of the observed signals and hence to better estimates θˆ. 2 A sensible initial estimate of the noise variance σℓ is simply the signal power 1 K−1 (ℓ) 2 K k=0 yk . We propose to use cyclic maximization (Stoica & Selen, 2004) to iteratively improve estimates σˆ2 of σ2. Speciﬁcally we alternate P ℓ ℓ ˆ 2 2 θ = argmax f y θ, σˆ1,..., σˆL (2.27) θ and

2 2 ˆ 2 2 σˆ1 ,..., σˆL = argmax f y θ, σ1 , . . . , σL . (2.28) 2 2 (σ1 ,...,σL )

The maximization (2.27) is the same as (2.23), while for (2.28) either the standard 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 our ← suﬃcient statistic −µSℓ ℓ=1,...,L and the signal powers. In the single wave case, the algorithm described by Eqs. (2.27) and (2.28) will be referred to as the ML method. In a second extension, we consider a scenario where M waves are linearly superposed. For any given frequency ω, the measured signals still can be modelled 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 Love waves exclusively, each of the M mappings takes the form of Eq. (2.17) or Eq. (2.18). It is straightforward to model all the waves simultaneously in the glue factor by using extended matrices Hℓ(θ1),..., Hℓ(θM ) for ℓ = 1,...,L ˜ T ˜ T T and an extended wave state s(θ1) ,..., s(θM ) . 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 setting M = 1 and estimating one wave as in the previous subsection. Assume now that we have estimated M wave parameter vectors. We increase M by one and iterate the following. Pick some m 1,...,M and update ˆ ∈ { } the estimate of θm while ﬁxing θj = θj j∈{1,...,M}\{m}. This amounts to changing the glue factor (2.21) to{ }

L 6m gθ (s ,..., s ) = δ s sˆ H (θ ) s˜(θ ) , (2.29) m 1 L ℓ − ℓ − ℓ m m ℓ=1 Y ˆ6m , H ˆ ˜ ˆ where sℓ j∈{1,...,M}\{m} ℓ(θj ) s(θj ) is the estimated state due to all the waves except for the m-th. The corresponding θˆ can again be found by P ← m maximizing over θm in Eq. (2.25), where −µS˜ is calculated using the glue factor (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 both synthetic and real data are shown. First the estimator mean-squared error (MSE) is compared with the CRB. Next, an example of the functioning of the ML method for modeling multiple waves is given using a simple monochromatic synthetic dataset. Then, a more sophisticated synthetic waveﬁeld is used to analyze the performance of the estimators presented in this paper. At last, a dataset acquired in a real seismic survey is considered.

Cramér-Rao Bound Analysis We are interested in comparing the MSE of the estimation for diﬀerent methods with the theoretical limit given by the CRB Kay (1993). We restrict ourselves to the analysis of the wavenumber κ as this is the parameter of most practical interest. For equal noise variance σ2 in all signals, the element of the Fisher information matrix corresponding to the wavenumber κ is

T 2 2 N ∂κ pn ∂2 ln f(y θ) α K n=1 ∂κ E | = . (2.30) − ∂κ2 P 2σ2 h i When sensors are arranged regularly spaced on a circle, the Fisher information matrix is diagonal. Therefore, the MSE of any unbiased estimator is lower- bounded as 2 E E 2 2σ (ˆκ [ˆκ]) 2 . (2.31) − ≥ N ∂κTp α2K n n=1 ∂κ P 25

We compare the MSE of the estimates from the ML method of the previous section and from the Bartlett beamformer (2.11) with the CRB by means of a numerical simulation. We consider an array of N = 7 sensors and a single Rayleigh wave with circular particle motion (i.e., ξ = π/4). In Fig. 2.6 the MSEs of the ML method and the Bartlett beamformer of Eq. (2.11) are compared with the CRB for different SNRs. We define SNR , α2/2σ2. The Bartlett beamformer, which uses only the vertical component, is outperformed by the ML method as expected. The ML method exhibits smaller MSE and achieves the CRB for a sufficiently large SNR. Even for high SNR the vertical component Bartlett beamformer does not achieve the CRB as the signals on the horizontal components are disregarded. At low SNR, the MSE saturates since the wavenumber estimate is constrained by the algorithm implementation to belong to a finite interval.

Monochromatic Seismic Wavefield We present a the first example, in which we generate a synthetic wavefield composed of two Rayleigh waves and two Love waves Maranò et al. (2011). The number of waves M = 4 is known. Waves are monochromatic at known frequency of 1 Hz. We use anarrayof 14 triaxial sensors, 500 samples, and 5 seconds of observation. Measurements are corrupted by additive white Gaussian noise, with different variance in each channel. Noise variances and wavefield parameters 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 models the presence of a single Rayleigh wave from iteration 1 until iteration 5. At iteration 6 the graph is enlarged to account for a second Rayleigh wave, therefore accounting for the simultaneous presence of two waves. At iterations 10 and 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 for an additional wave as the likelihood (not shown) converges to a stable value. 2 Similarly, Fig. 2.7(b) shows the estimates of noise variance σℓ . Sudden decrease in estimated variance in the graph corresponds to the inclusion of an additional wave in the graph. Fig. 2.8 depicts the log-likelihood function of a Love wave as a function of wavenumber κ and azimuth ψ in polar coordinates (κ, ψ). In Fig. 2.8(a) it is possible to see one strong peak at ψ3 = π/4 and no other strong peaks are visible. At iteration 14, one Love wave remains− in the wavefield and the associated peak, located at ψ4 = π, is now clearly visible, as shown in Fig. 2.8(b). At the 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 wavefield developed in the SESAME project Bard, P.-Y. (2008); Bonnefoy-Claudet et al. (2006a). This synthetic dataset captures the complexity of the seismic wavefield, accounting for the simultaneous presence of several seismic sources, emitting both short burst of energy and longer harmonic excitations. It is a wavefield 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 seismic sources in a structural model with one layer with low seismic velocities over a half-space with higher velocities. The body wave velocity profile for this model is depicted in Fig. 2.2(a). We use 38 triaxial sensors and solely 10 seconds of recording sampled at a rate of 100 Hz. Different frequencies are processed independently. 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 estimates of the wavenumber κ (black dots) suggest the Rayleigh wave dispersion curves. 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 ellipticity angle ξ for M = 1. Values of ξ ( π/2; 0) correspond to a retrograde particle motion 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 increased to M = 3. It is shown that increasing the number of waves modeled allows 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 and ellipticity angle of a Rayleigh wave are compared in the single wave setting. The Bartlett beamformer of Eq. (2.11) employing the sole vertical component, the three components Bartlett beamformer as in Eq. (2.14), and the ML method are compared. The estimates of the three components Bartlett beamformer and the ML method often belong to the higher modes while the vertical component Bartlett beamformer to the fundamental mode. This is probably due to a different distribution of energy among the horizontal and vertical components for the different modes. To see this compare the theoretical pattern of the Rayleigh wave ellipticity depicted in Fig. 2.11(b) and the Rayleigh wave described in Eq. (2.2) with the estimated wavenumbers among the modes in Fig. 2.11(a). The analysis of the sole vertical component does not allow the retrieval of Rayleigh wave ellipticity. Indeed, the elliptical motion cannot be inferred without considering the three components. The small difference between the estimates from the three component Bartlett beamformer and from the 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 real site of Brigerbad in Switzerland. The data was recorded during a seismic survey performed by the Swiss Seismological Service in 2010. The site is located in the Rhone valley, a deep alpine valley, in southern Switzerland. An array of twelve triaxial sensors is deployed in a flat area outside Brigerbad, with geometry as shown in Fig. 2.12. Every sensor has a GPS receiver collecting accurate timing information and thus enabling the synchronization of all the recordings. Concerning the physical placement of each sensor, a small hole is dug with the aim of removing the first layer of grass and earth. The horizontal alignment of each sensor is obtained with a spirit level and the alignment toward north with a 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 sampling frequency of 100 Hz. In Fig. 2.13, six signals from the Brigerbad measurement are shown. The signals depict 30 seconds of recording of the three components from sensor number 1, the central sensor, and sensor number 2. The two sensors are located 9.8 meters apart. Each trace depicts the velocity of the ground motion at the sensor position along a certain axis. Bursts of energy typical of ambient vibrations can be seen around 17 and 23 seconds. On the trace of the y component of the second sensor (fifth trace from the top), a strong harmonic component is particularly noticeable. This is occasionally found in urban envi- ronment where human activity can affect the measurements with narrowband contributions. The recorded signals are split in windows of the duration of ten seconds. Windows are processed independently as described in the earlier sections. We use the Parzen window method Duda et al. (2001) to obtain a single graphical representation 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 Rayleigh waves, 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 dispersion curves obtained with the ML method. The ML method models a fixed number of M = 3 waves. Compared to the Bartlett beamformer more outliers are present, 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. Only the estimates associated with the fundamental mode are shown. In Fig. 2.14(d), the estimates associated with the first higher mode are shown. The retrieval of the ellipticity curves provides additional information about the structure of the subsoil. Ellipticity curves can be used in addition to the dispersion curves in the formulation of the geophysical inverse problem. In most of the pictures, it is possible to see the impact of a narrowband interferer, presumably some industrial machinery such as a pump, at around 2.5 Hz. From such pictures we recognize the large number of outliers in real cases. However, a trained eye can identify the dispersion curves and the ellipticity curves.

2.6 Conclusions and Outlook

We have reviewed key aspects of the seismic wavefield and different types of waves with particular emphasis on surface waves and their properties. The most widely used array processing techniques to analyze seismic surface waves have been reviewed. After an introduction to factor graphs tailored to likelihood computation and state-space models, we have presented in more detail how it is possible to perform ML estimation of wavefield parameter of seismic surface waves. We have shown how to account for different noise variance on each channel, by properly merging the information from sensors with different noise level, and how to account for arbitrary sensor positions and arbitrary sampling instants. In the same framework, the superposition of multiple waves has also been addressed. We have compared the performances of different algorithms with theoretical limits. We have provided concrete examples from the analysis 28 of seismic surface waves. We have shown results from the ambient vibrations wave field, both from high-fidelity synthetics and from a real seismic survey. Further developments of the ML technique include the adaptive choice of the number of waves modeled. The goal is to find a trade off between goodness of fit and model complexity. We conjecture that this will remove many outliers. In the near future, modern sensors will enable to gather an increasing number of diverse measurement. An example is the quickly developing field of rotational seismology where sensors capable of measuring ground rotation are employed. Using the factor graph approach it is possible to readily adapts to this new type of measurement.

Acknowledgments

The authors wish to thank Dr. Jan Burjánek and Dr. Clotaire Michel for making the recordings of the Brigerbad survey available. This work is supported, in part, by the Swiss Commission for Technology and Innovation under project 9260.1 PFIW-IW and by Spectraseis AG. 29

0.05 0.05 ] ] 1 1 − −

[m 0 [m 0 ψ ψ sin sin κ κ

−0.05 −0.05 −0.05 0 0.05 −0.05 0 0.05 κmcos ψ [ −1] κmcos ψ [ −1] (a) 3 Hz (b) 4 Hz

0.05 0.05 ] ] 1 1 − − [m 0 [m 0 ψ ψ sin sin κ κ

−0.05 −0.05 −0.05 0 0.05 −0.05 0 0.05 κmcos ψ [ −1] κmcos ψ [ −1] (c) 5 Hz (d) 6 Hz

0.05 0.05 ] ] 1 1 − −

[m 0 [m 0 ψ ψ sin sin κ κ

−0.05 −0.05 −0.05 0 0.05 −0.05 0 0.05 κmcos ψ [ −1] κmcos ψ [ −1] (e) 7 Hz (f) 8 Hz

0.05

0.045 Fundamental mode

0.04 First higher mode

0.035 Second higher mode

0.03 Est. wavenumber κˆ

0.025

0.02

0.015

Wavenumber [1/m] 0.01

0.005

0 3 4 5 6 7 8 Frequency [Hz] (g) Wavenumbers estimated at diﬀerent frequencies.

Figure 2.4: The beamformer outputs P (ψ, κ) computed at diﬀerent frequencies are shown in Figs. 2.4(a)-2.4(f). Dark colors indicate high values while light colors indicate low values. The white crosses indicate the pairs (κ, ψ) maximizing the beamformer output. In Fig. 2.4(g) the estimated wavenumbers are plotted 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 codes as 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 variables into several factors. Each edge in the graph represents a variable and each node represents a factor. In our setup, the variables are random variables and the function a joint PDF of these variables. Consider the example of a hidden Markov model (HMM) with hidden variables X , (X0,..., XK ) and observable variables Y , (Y1,...,YK ). For such a model, the joint PDF f factors into conditional PDFs f as { k}k=0,...,K K f(x, y) = f (x ) f (x , y x ) . (S.1) 0 0 k k k | k−1 kY=1 Fig. S.1 depicts the corresponding factor graph. For such a model, the likelihood of the fixed observations Y = y is

f(y) = f(x, y) dx , (S.2) Zx 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 factorizing function f(x, y) due to the distributive law. This leads to the calculation of intermediate results, which can be viewed as messages being sent along the edges 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, we denote the forward message (the message sent in the same direction as the → edge) by a left-to-right arrow µ and the backward message (the message sent −Z ← in the opposite direction) by a right-to-left arrow −µZ .

f0 f1 fk fK X0 X1 Xk−1 Xk XK−1 XK

Y1 = y1 Yk = yk YK = yK

Figure S.1: Factor graph for a hidden Markov model. Consider a factor graph containing the generic factor g(s , . . . , s , z) de- →1 n picted in Fig. S.2. By deﬁnition, the sum-product message µ leaving this −Z → node on edge Z is calculated in terms of the incoming messages −µSj j=1,...,n and the local function g as { }

→ n → µ (z) , g(s , . . . , s , z) µ (s ) ds ds . (S.3) −Z ··· 1 n −Sj j 1 ··· n j=1 Zs1,...,snZ Y Using this message update rule, we can send messages in any cycle-free factor graph starting at the leaves of the graph, until we have calculated two messages → ← on 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 likelihood (S.2) can be calculated as

→ ← f(y) = −µXk (xk) −µXk (xk) dxk (S.4) Zxk

for any edge Xk representing some hidden variable in the factor graph. This applies 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 parameter vector θ aﬀecting only the factor f0:

K f(x, y θ) = f (x θ) f (y , x x ) . (S.5) | 0 0 | k k k | k−1 kY=1 In this case the likelihood function f(y θ) can be calculated as | → ← f(y θ) = µ (x θ) µ (x ) dx , (S.6) | −X0 0 | −X0 0 0 Zx0 ← where only the backward message µ depends on the data y. It follows −X0 ← immediately that, for ML estimation of θ, the message −µX0 is a suﬃcient statistic.

S → 1 µ−S g 1 Z → ← µ µ Sn −Z −Z → −µSn Figure S.2: Sum-product message passing through a generic factor g.

gθ (1) (1) (1) (1) gθ X0 Xk−1 Xk XK−1 ˜ ˜ , S1 S = s(θ) y(1) y(1) k K S1 = H1(θ)

(L) (L) (L) (L) X0 Xk−1 Xk XK−1 , S SL (L) (L) L HL(θ) yk yK

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

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 = Ak Xk−1 (S.7) Yk = c Xk + Zk for k = 1,...,K, where

cos Ωk sin Ωk Ak , − (S.8) "sin Ωk cos Ωk # is the state transition matrix with discrete-time frequency Ω , ω(t t ), k k − k−1 and c , (1, 0). In this model, ω is the ﬁxed continuous-time frequency, tk are 2 the (potentially non-uniform) sampling times, Xk R is the state vector, Zk ∈2 is zero-mean white Gaussian noise with variance σ , and Yk is the observable T output. If we ﬁx X0 = α(cos ϕ, sin ϕ) 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). A factor graph representation is given in Fig. S.1 with the internal factorization of a single factor fk being depicted in Fig. S.3. The four types of nodes appearing in 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 degenerate and scaled) multivariate Gaussian PDF. The forward message on some edge S → → m W is thus parameterized by a mean vector − S , an inverse covariance matrix←− S, ← W and a scale factor. For the backward message we use the notation m−S, −S. For the types of local functions listed in Table 2.1, the message update rules according to the sum-product rule (S.3) can be calculated explicitly in terms of the parameters of the messages. These update rules are derived and tabulated in (Loeliger et al., 2007).

fk Xk−1 Xk Ak =

c σ (0, 2) N Zk +

Yk = yk

Figure S.3: State-space model a for sinusoid. 33

10−3

10−4 ] 2 −

10−5 MSE [m

10−6 Bartlett Beamformer ML method Cramér-Rao Bound 10−7 −32 −30 −28 −26 −24 −22 −20 −18 −16 −14 SNR [dB] Figure 2.6: Comparison of the MSE of wavenumber estimates with the CRB at diﬀerent SNR. 34

0.9

0.8

0.7

0.6

0.5

0.4 Amplitude [m] αˆ1 0.3 αˆ2 0.2 αˆ3 0.1 αˆ4 0 2 4 6 8 10 12 14 16 18 Iteration (a) Estimated amplitude at diﬀerent iterations. The graph accounts for an additional wave at iteration 1, 6, 10, and 15. The dotted lines correspond to the true values.

0.9 ]

2 0.8

0.7

0.6 σˆ2 0.5 1 σˆ2 0.4 2 σˆ2 Noise Variance0.3 [m 3 2 σˆ4 0.2 2 σˆ5 0.1 2 σˆ6 0 2 4 6 8 10 12 14 16 18 Iteration (b) Estimated noise variance at diﬀerent iterations. Only six channels are shown. The dotted lines correspond to the true values.

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

0.06 0.06 0.06 ] ] ] 1 1 1 − − −

[m 0 [m 0 [m 0 ψ ψ ψ sin sin sin κ κ κ

−0.06 −0.06 −0.06 −0.06 0 0.06 −0.06 0 0.06 −0.06 0 0.06 κmcos ψ [ −1] κmcos ψ [ −1] κmcos ψ [ −1] (a) Iteration 1 (b) Iteration 14 (c) Iteration 18

Figure 2.8: Log-likelihood function of a Love wave. 35

0.07 Fundamental mode 0.06 First higher mode

0.05 Second higher mode Est. wavenumber κˆ 0.04

0.03

0.02 Wavenumber [1/m]

0.01

0 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz] (a) Rayleigh wave dispersion curve.

π 2

[rad] 4 ξ

- π

Ellipticity angle 4

Est. ellipticity ξˆ - π 2 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz] (b) Rayleigh wave ellipticity angle curve.

Figure 2.9: Estimates of κ and ξ obtained modeling a single (M = 1) Rayleigh wave. 36

0.07 Fundamental mode 0.06 First higher mode

0.05 Second higher mode Est. wavenumber κˆ 0.04

0.03

0.02 Wavenumber [1/m]

0.01

0 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz] (a) Rayleigh wave dispersion curve.

π 2

[rad] 4 ξ

- π

Ellipticity angle 4

Est. ellipticity ξˆ - π 2 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz] (b) Rayleigh wave ellipticity angle curve.

Figure 2.10: Estimates of κ and ξ obtained modeling three (M = 3) Rayleigh waves. 37

0.07 Vertical comp. 0.06 Three comp.

0.05 ML method

0.04

0.03

0.02 Wavenumber [1/m]

0.01

0 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz] (a) Rayleigh wave dispersion curve.

π 2

[rad] 4 ξ

- π

Ellipticity angle 4 Three comp. ML method - π 2 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz] (b) Rayleigh wave ellipticity angle curve.

Figure 2.11: Comparison of diﬀerent techniques for the analysis of a single Rayleigh wave. 38

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

x 0

m/s] −10

µ 10 0 −10 10 0 −10 10

x0 y −10 10

y0 z −10 10 Ground velocity [

z 0 −10 0 5 10 15 20 25 30 Time [s]

Figure 2.13: The picture shows six signals of duration 30 seconds from the Brigerbad measurement. The top three traces belong to sensor 1 and the bottom three to sensor 2 (cf. Fig. 2.12). 39

0.1 0.1

0.05 0.05 Wavenumber [1/m] Wavenumber [1/m]

0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Frequency [Hz] Frequency [Hz] (a) Love wave dispersion curve (ML method). (b) Rayleigh wave dispersion curve (ML method).

π π 2 2

π π [rad] 4 [rad] 4 ξ ξ

0 0

- π - π

Ellipticity angle 4 Ellipticity angle 4

- π - π 2 2 4 6 8 10 12 14 2 2 4 6 8 10 12 14 Frequency [Hz] Frequency [Hz] (c) Rayleigh wave ellipticity angle curve for (d) Rayleigh wave ellipticity angle curve for fundamental mode (ML method). the ﬁrst higher mode (ML method).

0.1 0.1

0.05 0.05 Wavenumber [1/m] Wavenumber [1/m]

0 0 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Frequency [Hz] Frequency [Hz] (e) Love wave dispersion curve (Bartlett (f) Rayleigh wave dispersion curve (Bartlett beamformer, horizontal components). beamformer, vertical component).

Figure 2.14: Analysis of Love waves and Rayleigh waves at the Brigerbad site using 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 Donat Fäh1

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 local seismic hazard assessment, oil and gas prospection, and in geotechnical investigations. Array processing techniques are used in order to estimate wavefield properties such as dispersion curves of surface waves and ellipticity of Rayleigh waves. However, techniques presently in use often fail to properly merge information from three-components sensors and do not account for the presence of multiple waves. In this paper, a technique for maximum likelihood estimation of wavefield parameters including direction of propagation, velocity of Love waves and Rayleigh waves, and ellipticity of Rayleigh waves is described. This technique models jointly all the measurements and all the wavefield parameters. Furthermore it is possible to model the simultaneous presence of multiple waves. The performance of this technique is evaluated on a high-fidelity synthetic dataset and on real data. It is shown that the joint modeling of all the sensor components, decreases the variance of wavenumber estimates and allows the retrieval of the ellipticity value together with an estimate of the prograde/retrograde motion.

40 41

3.1 Introduction

Analysis of the seismic wavefield enables us to gather knowledge of geological and geophysical features of the subsoil. Indeed seismic wave attributes such as velocity of propagation or polarization reflect the properties of the structure in which the wave is propagating. The analysis of these properties allow geophysicists to gain insight into the subsoil avoiding more expensive invasive techniques (e.g., borehole techniques). Seismic surveying methods represent a valuable tool in oil and gas prospection (Sheriff & Geldart, 1995) and in geophysical investigations (Tokimatsu, 1997; Okada, 1997). In this paper we present an application to ambient vibrations of a recently proposed 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. The advantage of passive methods is their applicability to urbanized areas and the ability to analyze lower frequencies that cannot be excited with active techniques, thus allowing to resolve deeper structures in the earth (Okada, 1997). In the case of a structure with low-velocity sediments above rock, the seismic wavefield of ambient vibrations is primarily composed of surface waves. But other waves are present such as body waves and resonances. In particular, the seismic wavefield is composed of an unknown number of simultaneously present waves 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 and wave polarization. Specifically, we are interested in retrieving the dispersion relation for both Love wave and Rayleigh wave, and Rayleigh wave ellipticity. In order to infer subsurface features of the earth, it is necessary to solve a geophysical inverse problem, see e.g. (Tarantola, 2004). The properties of the seismic wavefield deduced from seismic surveys are used in such an inverse problem. For the analysis of surface waves from ambient vibrations, a planar sensor array is typically deployed and array processing techniques are employed. Most of the array processing techniques in use assume planar wave fronts. In particular, frequency-domain beamforming techniques can be used. Central to these methods is the estimation of the spectral spatial covariance matrix, see e.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 summed up. Delays are computed as a function of angle of arrival and velocity of propagation. The final estimates of these two parameters are the values that maximize the sum. In the Capon method the sum is weighted by complex gains in order to reduce the impact of noise and the disturbance from interfering signals. Another technique that found application in the seismological community is the MUSIC algorithm (Schmidt, 1986). In this method an eigendecomposition of the spectral spatial covariance matrix is performed and properties of the noise subspace are exploited. Cornou et al. (2003a) have used the MUSIC algorithm for the analysis of ambient vibrations. 42

Single station approaches employing the three components of a triaxial seismometer exist (Christoffersson et al., 1988). In array processing, however, for a long time only the vertical component has been used. In recent work (Fäh et al., 2008), a technique was proposed to analyze also the horizontal components. The technique allows to distinguishing between Love waves and Rayleigh waves. Vertical and horizontal components are however processed separately, leading to sub optimal performances. Further work proposes a method for the estimation of Rayleigh wave ellipticity (Poggi & Fäh, 2010). Also this latter work lacks of a joint treatment of all the three components. In this paper, we describe a recently developed technique to perform ML parameter estimation of wave parameters (Maranò et al., 2011). This technique models jointly the measurements from all components and all the parameters. It will be shown that this leads to a substantial improvement in the retrieval of the dispersion curves. In addition, an estimate of Rayleigh wave ellipticity including the sense of rotation of the particle is provided. We believe that this new information will provide a valuable additional constraint for the geophysical inversion. The technique also allows to address the issue of multiple waves by means of wave field decomposition within the same framework leading to a more accurate parameter estimation and the detection of weaker waves. We assess the performance of the proposed technique on the ambient vibrations wave field, both on high-fidelity synthetics and on real data, and compare with classical beamforming (Bartlett method). The paper is organized as follows. In Section 2, we introduce our notation thereby recalling the wave equations of the displacement field induced by Love waves and Rayleigh waves, and we elaborate on the representation of Rayleigh wave ellipticity. In the same section we also define the estimation problem addressed in this paper. In Section 3 we present the technique central to this paper emphasizing its novel contributions. In Section 4 we provide numerical examples of the analysis of ambient vibrations from both high fidelity synthetics and 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 the surface of the earth. We restrict our interest to small aperture arrays and work with a flat earth model. We use a three-dimensional, right-handed Cartesian coordinate system with the z axis pointing upward. The azimuth ψ is measured counterclockwise from the x axis. Each sensor measures the ground velocity along the direction of the axes of the coordinate system x, y, and z. We say that each sensor has three components, each component measuring the motion of the ground along a certain direction. For the sake of simplicity, we provide wave equations of the displacement field u, despite the actual measurement is ∂u R3 the velocity field ∂t .The displacement field, at position p and time t can be described by the vector field ∈ u(p, t) , (u (p, t), u (p, t), u (p, t)) : R4 R3 . x y z → In this paper, we study waves propagating near the surface of the earth and 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 Love waves and Rayleigh waves. The wave equations we describe hereafter are valid for z = 0 and for plane wave fronts. The direction of propagation of a wave is given by the wave vector κ = κ (cos ψ, sin ψ, 0)T, whose length κ is the wavenumber. Love waves exhibit a particle motion conﬁned to the horizontal plane. The particle oscillates perpendicular to the direction of propagation. The particle displacement generated by a single Love wave at position and time (p, t) is

u (p, t) = α sin ψ cos(ωt κTp + ϕ) x − − u (p, t) = α cos ψ cos(ωt κTp + ϕ) (3.1) y − uz(p, t) = 0 .

u (p, t) = α sin ξ cos ψ cos(ωt κTp + ϕ) x − u (p, t) = α sin ξ sin ψ cos(ωt κTp + ϕ) (3.2) y − u (p, t) = α cos ξ cos(ωt κTp + π/2 + ϕ) . z − We call ξ [ π/2, π/2] ellipticity angle of the Rayleigh wave. This quantity determines∈ the− eccentricity 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 oscillation on the vertical component uz is shifted by +π/2 radians with respect to the oscillation on the direction of propagation). If ξ (0, π/2) the wave is said to be prograde. For ξ = 0 and ξ = π/2 the polarization∈ is vertical and horizontal, respectively. The quantity tan±ξ is known as the ellipticity of the Rayleigh wave. | | We now explain in more detail the parametrization of ellipticity used to model Rayleigh waves as used in Eq. 3.2. Commonly, Rayleigh wave ellipticity is referred to as the ratio of the absolute values of the amplitude on the radial component and on the vertical component, i.e., the H/V ratio. Considering equation Eq. (3.2), and defining H= α sin ξ and V = α cos ξ it follows that | | | | H α sin ξ = | | = tan ξ . V α cos ξ | | | | Note that there is no information about the sense of rotation of the particle in the H/V ratio as the sign of tan ξ is lost. By considering directly the ellipticity angle ξ it is possible to preserve this information and infer the sense of particle rotation. Fig. 3.1 depicts the two different representations for Rayleigh wave ellipticity in the case of a layer over a half space and clarifies this idea. Namely, the SESAME 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 motion of the fundamental mode is retrograde at low frequencies (Malischewsky et al., 2008, 2006). At each singularity (i.e., H = 0 or V = 0) the sense of rotation 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 25 Layer 2 2000 1000 100 50 2500 ∞

100 + π 2

10 + π

V 4 [rad] / ξ

1 0 Ellipticity H 0.1 Fundamental - π Fundamental 4 First higher First higher Second higher Ellipticity angle Second higher Third higher Third higher 0.01 - π 2 4 6 8 10 2 2 4 6 8 10 Frequency [Hz] Frequency [Hz] (a) Rayleigh wave ellipticity curve in the (b) Rayleigh wave ellipticity curve in the el- common H/V representation. lipticity angle ξ representation.

Figure 3.1: Two diﬀerent representations of Rayleigh wave ellipticity in a layer over 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 occurs and the particle motion is horizontally polarized. Between 2 Hz and 3.8 Hz the 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 information about the sense of rotation of the particle and we are able to draw the above conclusions only because of our knowledge about the structural model. In Fig. 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 when ξ ( π/2, 0) and it is prograde when ξ (0, π/2). The polarization is vertical for∈ξ −= 0 and horizontal for ξ = π/2. Similar∈ considerations can be made for the higher modes. This latter representation± of Rayleigh wave ellipticity allows to visualize the sense of rotation of the wave.

Problem Statement Our end goal lies in the estimation of wavefield parameters θ based on noisy measurements y from an array of seismometers. For a Love wave we define the 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 the measurements y given a specific parameter vector θ. Second, these likelihood| computations 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 the accuracy of the parameter estimation. In addition, we assume the noise variance to be different on each sensor and on each sensor component. Therefore we are interested in estimating these noise variances. In the estimation of wavefield parameters, more weight is given to 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 field thereby tackling the superposition of an unknown number of waves of different type. 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 and a long recording is split in shorter time windows. The composition of the wavefield is allowed to change at different frequencies and in different time windows. In this section, we describe the modeling of multiple monochromatic waves with the same frequency. The wavefield composition (i.e., the number and the type of waves) is assumed to remain unchanged within each time window. An informal high-level description of the operating principle of the proposed method is provided in Algorithm 1.

Maximum Likelihood Parameter Estimation Our interest lies in computing the likelihood of the observations y for a speciﬁc wave type and wave parameter vector θ. Then a maximization of the likelihood function 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 (ℓ) measurements Yk at discrete instants tk for k = 1,...,K are (ℓ) (ℓ) Yk = u(pℓ, tk) + Zk , where u(pℓ, tk) is a deterministic function with unknown waveﬁeld parameters (ℓ) 2 θ and Zk is zero-mean additive white Gaussian noise with variance σℓ . With this signal model, the PDF of the observations y is

L K 2 1 − y(ℓ)−u(p ,t ) /2σ2 p(y θ) = e k ℓ k ℓ , (3.3) 2 | 2πσℓ ℓY=1 kY=1 p 46 where we have grouped all the measurement as y = y(ℓ) k=1,...,K. { k } l=1,...L Observe that, for a given wave type, u(pℓ, tk) is a deterministic function of the waveﬁeld parameters θ for each ℓ and k. This function is written explicitly in Eqs. (3.1) and (3.2). The ML estimate θˆ of the parameter vector θ is obtained by means of the following maximization

θˆ = argmax p y θ . θ This suﬃces to estimate wave parameters in the single wave setting. More details on ML estimation can be found, e.g., in Kay (1993).

Wavefield Decomposition In the seismic wavefield several waves of different functional form are present simultaneously. This superposition can severely downgrade the quality of the estimation process if not addressed appropriately. In this work, we propose an approach, called wavefield decomposition, enabling us to separate the contribution of different waves and improving the accuracy of the parameter estimation. Assuming a linear medium, each sensor records the linear superposition of such waves. Therefore, in presence of M waves, we have that the measurement (ℓ) Yk is M (ℓ) (m) Yk = u (pℓ, tk) + Zk , m=1 X (m) where u (pℓ, tk) is the contribution of the m-th wave. It follows immediately, that in the multiple wave setting, the PDF (3.3) M (m) should be altered by replacing u(pℓ, tk) by m=1 u (pℓ, tk). The PDF is now parametrized by several wavefield parameter vectors, (θ1,..., θM ). In principle, also in the multiple wave settingP it is possible to obtain wave parameter estimates by maximizing

θˆ1,..., θˆM = argmax p(y θ1,..., θM ) . (θ1,...,θM ) | Unfortunately, such a maximization is unfeasible, even for small M, because the parameter space is increased M-fold. Therefore we propose a greedy algorithm that increases gradually the number of waves modeled. The algorithm begins modeling a single wave and estimates the parameter vector θ1 of the first wave. This wave can be either a Love wave or a Rayleigh wave. In a second step, the parameters of the first wave are kept fixed to θˆ1 while the maximization is performed over θ2. The number of waves modeled by the algorithm is increased gradually until a stopping criterion is reached. Each estimated parameter vector benefits from the estimation of the other waves as the parameter estimation is repeated iteratively.

Model Selection Two questions arising naturally are how to choose the wave type and how many 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 modeling additional waves. Considering a set of possible models, diﬀering for wave type and number of waves, the model with the smallest BIC is selected. The BIC is deﬁned as BIC = 2p y θˆ ,..., θˆ + N ln(LK) , − 1 M p where Np denotes the total number of estimated parameters of the model and LK is the number of measurements. In order to limit the computational complexity of the proposed method, we set 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 noise with zero mean. However, we do not assume the noise variance to be equal in different sensors or components. The estimation algorithm properly weights measurements from channels with different noise level. An ML estimate of noise variance can be obtained with the following maximization 2 2 ˆ ˆ 2 2 σˆ1 ,..., σˆL = argmax p y θ1,..., θM , σ1 , . . . , σL , 2 2 (σ1 ,...,σL ) where the wavefield parameters are kept fixed and the maximization is per- 2 formed only on the σℓ ℓ=1,...,L. Because of the signal model, it is equivalent { } 2 to perform L separate maximizations on σℓ for ℓ = 1,...,L. Since the wavefield parameter estimates are influenced by the different noise variances we iteratively repeat the two maximizations 2 2 ˆ ˆ 2 2 σˆ1 ,..., σˆL = argmax p y θ1,..., θM , σ1, . . . , σL 2 2 (σ1 ,...,σL ) and ˆ ˆ 2 2 θ1,..., θM = argmax p y θ1,..., θM , σˆ1 ,..., σˆL . (θ1,...,θM ) Being the likelihood a finite value, this iterative maximization is guaranteed to converge. An initial estimate for the noise variance can be obtained from the signal energy K 1 σˆ2 = (y(ℓ))2 . ℓ K k Xk=1 Additional Details The description of this section provides a rigorous description of the functioning of the proposed method and makes an implementation of the method possible using tools widely used in statistics. However, in our implementation, instead of computing (3.3) directly, we model the PDF of the observations by means of a factor graph (Loeliger et al., 2007). The factor graph formalism allows to to derive a sufficient statistic and enables us to perform ML parameter estimation in a computationally attractive manner. Further details of our implementation relying on factor graphs are given in 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. ← 2 {Initial estimate for σℓ :} 2. for ℓ = 1 to L do 2 1 K (ℓ) 2 3. σˆℓ = K k=1(yk ) 4. end for P {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 ˆ(T) ˆ ˆ (T) 2 2 9. θm = argmaxθm p(y θ1,..., θm−1, θm , σˆ1 ,..., σˆL) 2 2 | ˆ ˆ 2 2 10. σˆ1 ,..., σˆL = argmax σ2,...,σ2 p(y θ1,..., θm, σ1 , . . . , σL) ( 1 L ) | ˆ ˆ 2 2 11. until convergence of p(y θ1,..., θm, σˆ1 ,..., σˆ ). | L 12. Compute BIC for a model of m waves. 13. end for 14. Choose model with smallest BIC. Potentially, stop adding waves and exit. {Reﬁne estimation of existing waves:} 15. repeat 16. for i = 1 to m do 17. θˆ = argmax p(y θˆ ,..., θˆ , θ , θˆ ,..., θˆ , σˆ2,..., σˆ2 ) i θi | 1 i−1 i i+1 m 1 L 18. end for 2 2 ˆ ˆ 2 2 19. σˆ1 ,..., σˆL = argmax σ2,...,σ2 p(y θ1,..., θm, σ1 , . . . , σL) ( 1 L ) | ˆ ˆ 2 2 20. until convergence of p(y θ1,..., θm, σˆ1 ,..., σˆ ). | L 21. end for

Summary of Contributions The proposed method brings several improvements with respect to techniques currently in use. The proposed technique enables us to perform ML parameter estimation • of wavefield parameter in a monochromatic wavefield relying on measurements corrupted by additive white Gaussian noise. The approach accounts for all the measurements and all the parameters jointly. Ap- plicability of the proposed technique is not limited to the application presented in this paper. In particular, the technique allows to combine measurements from different types of sensors, and is readily extensible to waves with different polarization and to spherical wave fronts. The technique 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 separation and in the identification of singularities of the ellipticity (i.e., peaks and minima 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 accuracy of each wave increases as parameters are iteratively re-estimated. This leads to the decomposition of the wavefield and allows the detection of weaker waves. The proposed technique estimates the noise variance in each channel. • This brings about various advantages. It enables us to use sensors of different technology and therefore with different noise levels. A misplaced or badly working sensor, will exhibit a higher noise level and will be automatically given less weight in the estimation process. Alternatively, it is possible to identify sensors having suspiciously high noise variance and perform a target check on that specific sensor. The issues of spatial sampling and array geometry are outside the scope • of this work. However, it is known that the joint usage of all the sensor components 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 increasing complexity. First, in Section 3.4, we compare the MSE of the proposed estimator with the CRB and the MSE of other estimators. In Section 3.4, we analyze a synthetic monochromatic wavefield with the aim of demonstrating the functioning of the algorithm in detail. In Section 3.4, we assess the performance of the algorithm on high-fidelity synthetics of the ambient vibrations wavefield developed 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 sites in Switzerland are presented. The data was recorded during seismic surveys performed by the Swiss Seismological Service in 2011. We now give some details about the processing. All frequencies are processed independently. We apply no filtering to the recordings other than mean removal. The whole signal is split into blocks (time windows) of equal length within which the signal is assumed to be stationary. For comparison, we present results obtained using the three-components method for vertical, radial, and transverse component proposed in Fäh et al. (2008) using the same window length. We will refer to the three-components technique simply as “classical beamforming”. In the figures, dispersion curves are shown in wavenumber (in m−1), versus frequency (in Hz). Ellipticity curves are shown both in ellipticity H/V and in ellipticity angle ξ versus frequency. From the processing of long recordings, a large quantity of estimated wave parameters is available. In order to obtain a single picture representative of the results from the whole recording, we use the Parzen window method (Duda et al., 2001). The resulting gray-scale pictures depict with darker color parameter 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 distance (dmin and dmax respectively), the minimum and maximum resolvable 50 wavenumber are defined as 2π π κmin = and κmax = . dmax 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 theoretical limit given by the CRB (Kay, 1993). The CRB is a lower bound on the variance of unbiased estimators. We restrict ourselves to the analysis of the wavenumber κ as this is the parameter of most practical interest. For equal noise variance σ2 in all signals, the element of the Fisher information matrix corresponding to the wavenumber κ is

T 2 2 N ∂κ pn ∂2 ln p(y θ) α K n=1 ∂κ E | = . (3.4) − ∂κ2 P 2σ2 h i When sensors are arranged regularly spaced on a circle, the Fisher information matrix is diagonal. Therefore, the MSE of any unbiased estimator is lower- bounded as 2 E E 2 2σ (ˆκ [ˆκ]) 2 . (3.5) − ≥ N ∂κTp α2K n n=1 ∂κ We compare the MSE of three differentP estimators with the CRB by means of a numerical simulation. We consider the vertical and the radial component beamforming of Fäh et al. (2008) and the ML method of Section 3. We consider an uniform circular array of N = 7 sensors and a single Rayleigh wave with elliptic particle motion defined by ξ = π/3. Such a wave has most of the energy on the horizontal components. In Fig. 3.2 the MSEs of the ML method and the classical beamforming are compared 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 constrained by the algorithm implementation to belong to a finite interval. As the SNR increases, the ML method always exhibits smaller MSE. For sufficiently large SNR, the ML method achieves the CRB. Even for high SNR the vertical component beamformer and the radial component beamformer do not achieve the CRB as they disregard the energy on the horizontal components or on the vertical component. The radial component beamformer exhibits in general smaller MSE than the vertical component beamformer because most of 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 Love waves and two Rayleigh waves. All waves are monochromatic with known frequency of 1 Hz. We use an array of 14 triaxial sensors, 500 samples, and 5 seconds of observation. The measurements are corrupted by additive white Gaussian noise, with different variance in each channel. The true wave field 51

10−4 ] 2 −

10−5 MSE [m

10−6 Vertical component Radial component

ML method 10−7 Cramér-Rao Bound

−26 −24 −22 −20 −18 −16 −14 −12 −10 −8 SNR [dB] Figure 3.2: Comparison of the MSE of wavenumber estimates with the CRB at diﬀerent SNR.

(R) T (R) π T parameters are θ1 = (0.9, 0, 0.03, π/4, π/4) , θ2 = (0.7, 4 , 0.03, π/2, π/4) , (L) π T (L) T θ3 = (0.8, 3 , 0.04, π/4) , and θ4 = (0.2, π, 0.04, π) . The noise variances, the wavefield parameters,− and the number and type of waves are unknown to the algorithm. Fig. 3.3(a) shows how the estimates of the amplitudes αℓ converge toward their true values (dotted lines) after a sufficient number of iterations. The algorithm 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- 2 timates of the noise variances σℓ . Sudden changes in estimated variance in the graph correspond to the inclusion of an additional wave in the graph. The improvements in the estimated parameters between two wave inclusions, are due 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 function of wavenumber and azimuth. At iteration 1, the algorithm computes the likelihood function for Love waves and Rayleigh waves, as seen in the two leftmost columns. Two strong peaks are visible for Rayleigh waves, at azimuths π/4 and π/2. For Love waves, only one peak at azimuth π/4 is visible. The algorithm chooses to model, as first wave m = 1, a Rayleigh− wave. At iteration 6, the first two columns are again showing the likelihood of the data for Love waves and Rayleigh waves also modeling the Rayleigh wave previously estimated. For Rayleigh waves, now only a single peak is visible as the contribution from the first wave is already modeled. The depiction of the likelihood function for Love waves appears to be substantially unchanged. The second wave modeled by the algorithm is a Love wave. At iteration 11, an additional Rayleigh wave is modeled. At iteration 14, only one Love wave remains in the wavefield (the (L) wave parametrized by θ4 ) 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 using the BIC. 52

0.9

0.8

0.7

0.6

0.5

0.4 Amplitude [m] αˆ(R) 0.3 1 αˆ(L) 0.2 3 (R) αˆ2 0.1 (L) αˆ4 0 2 4 6 8 10 12 14 16 18 Iteration (a) Estimated amplitudes at diﬀerent iterations. The graph accounts for an additional wave at iteration 1, 6, 11, and 14. The dotted lines show the true amplitude of the waves.

0.9

0.8 ] 2 0.7

0.6 2 0.5 σˆ1 σˆ2 0.4 2 σˆ2 0.3 3

Noise Variance [m σˆ2 0.2 4 σˆ2 0.1 5 2 σˆ6 0 2 4 6 8 10 12 14 16 18 Iteration (b) Estimated noise variances at diﬀerent iterations. Only six channels are shown. 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 layer over a half-space developed during the SESAME project (Bonnefoy-Claudet et 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, each window is processed independently. An array of 14 sensors, with an aperture of roughly 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 M2.1 of the SESAME dataset). In the ﬁgures, the results for detected waves are overlaid with the theoretical dispersion curves and ellipticity curves computed from the structural model parameters in Table 3.1. For both dispersion and ellipticity curves, the red solid line refers to the fundamental mode, the dashed blue line to the ﬁrst 53 ) ) 4 4 θ θ , , 3 3 ˆ ˆ θ θ , , (Love) 2 2 ˆ ˆ θ θ , , 1 1 = 4 ˆ ˆ θ θ | | y y ( ( m p p ) ) 4 4 ) ˆ ˆ θ θ 3 , , θ 3 3 , θ θ 2 , , ˆ θ 2 2 , ˆ ˆ θ θ 1 (Rayleigh) , , ˆ θ 1 1 | ˆ ˆ y θ θ | | ( y y = 3 p ( ( p p m ) ) 4 4 ) ˆ ˆ θ θ 3 ) , , ˆ θ 2 3 3 , ˆ ˆ θ θ θ 2 , , , (Love) θ 1 2 2 , ˆ θ θ θ 1 | , , ˆ y θ 1 1 | ( = 2 ˆ ˆ θ θ y | | p ( y y p ( ( m p p ) ) 4 4 ) ˆ ˆ θ θ 3 ) , , ˆ θ 2 3 3 ) , ˆ ˆ ˆ θ θ θ 1 2 , , , ˆ θ θ 1 2 2 | , ˆ ˆ θ θ θ y 1 (Rayleigh) | , , ( θ y 1 1 | p ( y θ θ | | p ( y y = 1 p ( ( p p m ) 4 ) ˆ θ 3 ) , 4 4 4 4 4 ˆ θ 2 3 ) , ˆ ˆ θ θ 10 10 10 10 10 1 2 ) , , ˆ ˆ θ θ ) × × 1 × × × 2 , , R ˆ ˆ θ θ ) 1 ( , , R ˆ θ ) 1 θ 587 185 583 581 557 ( , | . . . . . R ˆ θ ) y θ ( , | ( R ) θ y ( | p =4 =4 =3 =3 =3 ( R θ y ( | p ( y θ | p ( BIC BIC BIC BIC BIC y p LL Rayleigh wave ( p ) 4 ) ˆ θ 3 ) , 4 4 4 4 4 ˆ θ 2 3 ) , ˆ ˆ 10 10 10 10 10 θ θ 1 2 ) , , ˆ ˆ θ θ ) × × × × × 1 2 , , L ˆ ˆ θ θ ) 1 ( , , L ˆ θ ) θ 1 688 ( 123 037 553 555 | , . . . . . L ˆ θ ) θ y ( | , ( L ) θ y ( | p =4 =4 =4 =3 =3 ( L y θ ( | p ( θ y | p LL Love Wave ( y BIC BIC BIC BIC BIC p ( p 1 6 11 14 18 It.

Figure 3.4: The normalized LL functions for Love waves and Rayleigh waves at different stages of the algorithm are depicted in polar coordinates as a function of wavenumber and azimuth (in each picture, the horizontal axis is κ cos ψ and the vertical axis is κ sin ψ). The two leftmost columns show the LL of the residual wavefield, i.e. the LL function of an additional wave while the parameters of the waves estimated in previous iterations are kept fixed. The other columns show the LL for the waves modeled by the algorithm. The BIC values shown motivates the choice of wave type and the termination of the algorithm. 54

2080

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Figure 3.5: Rayleigh wave and Love wave dispersion curves obtained using the method in (Fäh et al., 2008) for the model M2.1. higher mode, the dot-dashed magenta line to the second higher mode, and the dotted green line to the third higher mode. Theoretical curves for Rayleigh wave and Love wave modes are depicted with the same colors but never appear in the same picture. Fig. 3.5 depicts results of the method in (Fäh et al., 2008). The Rayleigh wave 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 transverse component (Fig. 3.5(d)). Fig. 3.6 depicts Love wave and Rayleigh wave dispersion curves as estimated with the ML technique. Fig. 3.6(a) and 3.6(c) refer to the algorithm modeling at most one wave (Mmax = 1). Fig. 3.6(b) and 3.6(d) refer to the joint modeling of at most three waves (Mmax = 3). In general, we observe that the wavenumber estimates exhibit less scatter and less outliers when compared with the results depicted in Fig. 3.5. This is due 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

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(a) Rayleigh wave dispersion curve, Mmax = (b) Rayleigh wave dispersion curve, Mmax = 1. 3.

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Figure 3.6: Rayleigh and Love wave dispersion curves obtained using the ML technique 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 these four figures all the estimated parameters are plotted and contribution of different modes are not distinguished as easily as for dispersion curves. For this reason, estimates corresponding to the first mode are isolated in the frequency- wavenumber plane and only the corresponding ellipticity estimates are shown in Fig. 3.8. In this latter figure the behavior of the ellipticity of the fundamental mode can be understood more clearly. In in Fig. 3.8(c) and 3.8(d) it is possible to 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 is possible to appreciate how the modeling of multiple waves makes it easier to follow the curves. Also, the estimated curve just above the resonance frequency of 2 Hz appears more accurate. In Fig. 3.9 we compare the estimated wavenumbers for Rayleigh waves at different frequencies using different methods. Each figure shows the wavenumber estimates at a fixed frequency. The pictures in 3.9 can be compared with Fig. 3.5(b), 3.5(c), 3.6(a), and 3.6(b). The theoretical wavenumbers are shown with 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 is substantially undetected by the vertical beamforming and detected with some 56

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(a) Rayleigh wave ellipticity curve, Mmax = (b) Rayleigh wave ellipticity curve, Mmax = 1. 3.

π π 2 2

π π [rad] 4 [rad] 4 ξ ξ

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- π - π 2 1 2 3 4 5 6 7 8 9 10 11 2 1 2 3 4 5 6 7 8 9 10 11 Frequency [Hz] Frequency [Hz] (c) Rayleigh wave ellipticity angle curve, (d) Rayleigh wave ellipticity angle curve, Mmax = 1. Mmax = 3. Figure 3.7: Rayleigh wave ellipticity curves obtained using the ML technique for the 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 method detects both the fundamental and the ﬁrst higher mode. The two modes are detected separately by the vertical and the radial beamforming due to the different ellipticity of the diﬀerent modes. At 8.0 Hz, (Fig. 3.9(c)) the fundamental mode 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 the estimators. Indeed the estimated mode might be a mixture of the second and the third higher mode. In general, the proposed method also exhibit a smaller amount of outliers.

Brigerbad, Wallis The Brigerbad site is located in the Rhone valley, a deep Alpine valley, in southern Switzerland. An array of 12 Lennartz 5 seconds triaxial sensors is used. The layout of the array is depicted in Fig. 3.10(a). The whole recording is 58 minutes long and it is split into 10 seconds windows which are processed independently. Sampling rate is 200 Hz. 57

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Figure 3.8: Rayleigh wave ellipticity curves obtained using the ML technique for 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 method in Fäh et al. (2008). The fundamental mode of the Rayleigh wave is visible on the vertical component (Fig. 3.10(b)). The first higher mode of the Rayleigh wave is visible on the radial component (Fig. 3.10(c)) and only weakly on the vertical component. The fundamental mode of the Love wave is visible on the transverse component (Fig. 3.10(d)). Fig. 3.11 shows the results of the analysis performed using the ML technique described in this paper. Both the fundamental mode and the first higher mode of the Rayleigh wave are visible in Fig. 3.11(b). The fundamental mode of the Love wave is visible in Fig. 3.11(a). In Fig. 3.11 Rayleigh wave ellipticity curves are shown for different modes and different representations. The ellipticity of the fundamental mode is shown in 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 angle representation of Fig. 3.11(d). Analogously, in Fig. 3.11(e) and 3.11(f) the Rayleigh wave ellipticity for the first higher mode is shown. 58

Vertical Vertical Radial Radial ML (1) ML (1) ML (3) ML (3)

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

Rheintal, St. Gallen The Rheintal site is located in the Rhein valley, an Alpine valley, in eastern Switzerland. An array of 13 Lennartz 5 seconds triaxial sensors is used. The layout of the array is depicted in Fig. 3.12(a). The whole recording is almost six hours long and it is split in 10 seconds windows which are processed independently. Sampling rate is 200 Hz. Fig. 3.12 shows the results of the analysis performed using the method in Fäh et al. (2008). The fundamental mode of the Rayleigh wave is visible only in the vertical component (Fig. 3.12(b)). The Love wave fundamental mode is weakly visible on the transverse component. The analysis of the radial component brings no clear information. Fig. 3.13 shows the results of the analysis performed using the ML technique described in this paper. The fundamental mode of the Rayleigh wave is visible in Fig. 3.13(b). The fundamental mode of the Love wave is visible in Fig. 3.13(a). We note that the Love wave dispersion curve is now clearly visible with the ML method. Since the ML technique chooses between Love and Rayleigh wave adaptively, the algorithm tends to model the stronger waves 59

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

first, then removes its contribution, allowing the detection of weaker signals (in this instance the fundamental mode Love wave), and the final elaboration is improved. In Fig. 3.13(c) and 3.13(d) Rayleigh wave ellipticity curves of the fundamental mode are shown in the different representations. We emphasize how the zero of the H/V curve around 2.5 Hz is again clearly identified by looking at 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 waves from ambient vibrations recording of a recently developed technique for array processing of the seismic waveﬁeld. The technique performs ML waveﬁeld parameter estimation accounting for all the measurements and all the parameters jointly. The technique allows to model the simultaneous presence of multiple waves. Notably, we provide 60

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Figure 3.11: Dispersion curves and ellipticity curves obtained using the ML technique for Brigerbad survey. These results are obtained from a single processing with Mmax = 3. 61

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Figure 3.12: Rayleigh wave and Love wave dispersion curves obtained using the 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 synthetic dataset from the SESAME project and on real data from two surveys. This method improves estimates of Rayleigh and Love waves dispersion curves, and allows for an estimate of Rayleigh wave ellipticity. We have also shown that modeling multiple waves enables us to detect weaker waves that are not visible with traditional methods. Further developments of the method will include an adaptive window selection 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 invaluable assistance during the Rheintal measurement campaign. We also wish to thank Spectraseis AG for providing technical support during the same survey. Con- 62

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0.01 - π 0.5 1 1.5 2 2.5 3 3.5 4 2 0.5 1 1.5 2 2.5 3 3.5 4 Frequency [Hz] Frequency [Hz] (c) Rayleigh wave ellipticity curve for funda- (d) Rayleigh wave ellipticity angle curve for mental mode. fundamental mode. Figure 3.13: Rayleigh wave and Love wave dispersion curves obtained using the ML technique for the Rheintal survey. Mmax = 3. cerning the Brigerbad dataset, the authors wish to thank Dr. J. Burjánek and Dr. C. Michel. This work is supported in part by the Swiss Commission for Technology 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 considerable attention in the last years. Recent advances in sensor technologies allow us 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 lack of instrumental observations. This is due to both the technical challenges involved in measuring rotational motions and to the widespread belief that rotational motions are insignificant. This paper addresses the joint processing of translational and rotational motions from both the theoretical and practical perspective. Our attention focuses on the analysis of motions of both Rayleigh waves and Love waves from recordings of single sensors and from an array of sensors. From the theoretical standpoint, analysis of Fisher information (FI) allows us to understand how the different measurement types contribute to the estimation of quantities of geophysical interest. In addition, we show how rotational measurements resolve ambiguity on parameter estimation in the single sensor setting. We quantify the achievable estimation accuracy by means of Cramér-Rao bound (CRB). From the practical standpoint, a method for the joint processing of rotational and translational recordings to perform maximum likelihood (ML) estimation is presented. The proposed technique estimates parameters of Love waves and Rayleigh waves from single sensor or array recordings. We support

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

4.1 Introduction

The joint analysis of translational and rotational motions has the potential to improve the estimation of important physical properties of the near-subsurface. The most striking feature of rotational motions is that, together with translational motions, they enable us to estimate velocity of propagation of seismic waves from a point measurement. The amount of rotational motion induced by a seismic wave is inversely proportional to the wavelength, and is thus related to the velocity of propagation. As a result, a six-components measurement of both translational and rotational motions at a single spatial location gather sufficient information to estimate the velocity of propagation of a seismic wave. This fact unleashes a myriad of potential applications (Igel et al., 2012; Lee et al., 2009). Different approaches to estimate or directly measure rotational motions have been developed in the past decades. In some early applications, ground rotations have been estimated from the spatial derivatives of the translational measurements from an array of sensors (Niazi, 1986; Oliveira & Bolt, 1989; Spudich et al., 1995). In Nigbor (1994), one of the earliest direct measurement of rotational motions, employing a solid-state rotational velocity sensor, is found. Recent advances in sensor technology allow us to directly measure rotational motions with unprecedented accuracy and/or portability (Schreiber et al., 2009; Nigbor et al., 2009; Lee et al., 2012). A technology used in modern and portable rotational sensors relies on an electrochemical transducer where the motion of a fluid caused by an external acceleration is converted into an electrical signal (Leugoud & Kharlamov, 2012). A comparison of array derived rotational motions with direct measurements is found in Suryanto et al. (2006). The direct measurement of rotational motions provides an additional independent observation of the seismic wavefield. This is extremely valuable, for instance, in the analysis of surface waves. The measurements of rotational motions supplement the measurements of translational motions, potentially increasing the accuracy of the estimation of the geophysical parameters of interest. Applications relying on the analysis of surface waves are numerous. Notably, the analysis of the seismic wavefield enables us to gather knowledge of geological and geophysical features of the subsoil. Indeed seismic wave attributes such as velocity of propagation or polarization reflect the properties of the structure in which the wave is propagating. The analysis of these properties allow geophysicists to gain insight into the subsoil and the assessment of local seismic 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 parameters from a single sensor is particularly desirable because of the simplicity of measurement operations. Concerning the analysis of translational motions, a well-established single station method is the H/V ratio technique which has been widely used for different purposes (Fäh et al., 2003; Bonnefoy-Claudet et al., 2006a). Other methods estimate Rayleigh wave ellipticity from a single station (Hobiger et al., 2009; Poggi et al., 2012). The processing of both 65 translational and rotational motions from a single sensor location has been also addressed and it has been shown that the retrieval of wave velocity is possible (Igel et al., 2005; Ferreira & Igel, 2009). When plane wavefronts can be assumed, array processing techniques are usually employed. The use of array processing techniques in seismology has a long history. The earliest techniques analyzed a single component (Capon, 1969; Lacoss et al., 1969). More recent developments allow us to separate Love waves and Rayleigh waves (Fäh et al., 2008) and to estimate Rayleigh wave ellipticity (Poggi & Fäh, 2010). Recent work from the authors, includes a ML estimation technique accounting for all measurements and wavefield parameters jointly (Maranò et al., 2012). To the best of our knowledge, at this time there are no applications to seismology of the joint processing of rotational and translational motions for an array of sensors. In this paper, we are interested in the joint analysis of translational and rotational motions induced by surface waves. We consider different aspects of the problem from a signal processing perspective. We investigate the potential and the limitations introduced through joint processing of these two types of measurement. A method exploiting all the available measurements is presented. Examples are provided to support the theoretical investigation and to show the applicability of the proposed method. One contribution of this paper is a method for the joint processing of translational motion and rotational motion recordings of surface waves. All the measurements are considered in a single framework and the algorithm provide an ML estimate of the wavefield parameters. We extend a method proposed by the authors in Maranò et al. (2012). The original method jointly accounts for the measurement from three components translational sensors and the wavefield parameters of Love waves and Rayleigh waves. The simultaneous presence of 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 each • statistical models of interest. Fisher information (FI) enables us to gain insight about the contribution of each measurement to the estimation of different wave parameters of interest. We show under which conditions, such as number of sensors, type of • sensors, and wave type, it is possible to identify wave parameters. We derive lower bounds on the achievable accuracy of the estimators for • the geophysical parameters of interest, namely wavenumber and ellipticity angle. This allows us to compare the performance of any algorithm with a an accuracy bound independent of estimation technique and algorithm implementation. We provide a comprehensive collection of numerical examples illustrating • the potential and limitations of the joint processing of translational and rotational motions. We show applications of the presented algorithm to two distinct real • datasets of the retrieval of Love wavenumber, Rayleigh wavenumber, and Rayleigh ellipticity angle for both single sensor and array measurements. 66

The remainder of the paper is organized as follows. Section 4.2 presents the system model. The notation used in this manuscript is introduced and Love wave and Rayleigh wave equations for translational and rotational motions are provided. In Section 4.3, we analyze from a theoretical standpoint the limitations and the performance improvement achievable by including the rotational motions in the processing. In Section 4.4, we present the algorithm used in this work for the estimation of waveﬁeld parameters. Finally, several numerical results are provided in Section 4.5, including examples on synthetic data and 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 its translational motions and its rotational motions. In this section, we introduce wave equations describing quantities of interest and a model of the measurements. We describe the seismic wavefield, at position p R3 and time t with the vector 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 translational motions and the last three the rotational motions. For the sake of simplicity, we portray wave equations of the seismic wavefield in displacements and rotations despite the actual measurements may be velocities or accelerations. We use a three-dimensional, right-handed Cartesian coordinate system with the z axis pointing upward. The azimuth ψ is measured counterclockwise from the 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 this section, we omit the dependence on (p, t) for conciseness of notation. From mechanics (Aki & Richards, 2002), rotational motions (ωx, ωy, ωz) are related to the curl of translational motions (ux, uy, uz) as 1 (ω , ω , ω ) = (u , u , u ) (4.2) x y z 2∇ × x y z xˆ yˆ zˆ ∂ ∂ ∂ = ∂x ∂y ∂z ,

u u u x y z

where xˆ, yˆ, zˆ are the versors of the coordinate system and where denotes the determinant of the matrix. It follows directly from (4.2) that |·| 67

1 ∂u ∂u ω = z y x 2 ∂y − ∂z 1 ∂u ∂u ω = x z (4.3) y 2 ∂z − ∂x 1 ∂u ∂u ω = y x . z 2 ∂x − ∂y Boundary conditions at the free surface require all the stress along z to vanish and leave the displacement unconstrained. Let T denote the Cauchy stress tensor, we enforce the boundary conditions as Tzˆ = (0, 0, 0)T. Using Hooke’s law for a linear elastic medium, the following conditions are found at z = 0

∂u ∂u τ = µ z x = 0 x,z ∂x − ∂z ∂u ∂u τ = µ z y = 0 (4.4) y,z ∂y − ∂z ∂u ∂u ∂u ∂u ∂u τ = λ x + y + z + µ z z = 0 , z,z ∂x ∂y ∂z ∂z − ∂z where λ and µ are the Lamé parameters. Comparing (4.3) with (4.4) it is apparent that the conditions τx,z = 0 and τy,z = 0 are inﬂuencing the rotational motions. From (4.4) it is found that

∂u ∂u ∂u ∂u z = x and z = y , (4.5) ∂x ∂z ∂y ∂z substituting into (4.3) we ﬁnd the rotational motions at the free surface to be

∂u (p, t) ω (p, t) = z x ∂y ∂u (p, t) ω (p, t) = z (4.6) y − ∂x 1 ∂u (p, t) ∂u (p, t) ω (p, t) = y x . z 2 ∂x − ∂y Translational and Rotational Motions for Surface Waves In this paper, we study waves propagating near the surface of the earth and having a direction of propagation lying on the horizontal plane z = 0. The wave equations we describe hereafter are valid for z = 0 and for plane wave fronts. The direction of propagation of a wave is given by the wave vector κ = κ (cos ψ, sin ψ, 0)T, whose magnitude κ is the wavenumber. Love waves exhibit a translational particle motion conﬁned to the horizontal plane, 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 u (p, t) = α sin ψ cos(ωt κ p + ϕ) x − − · u (p, t) = α cos ψ cos(ωt κ p + ϕ) (4.7) y − · uz(p, t) = 0 , where α R+ and ϕ denote the amplitude and the phase of the wave, respectively. The∈ temporal angular frequency is denoted with ω. The azimuth ψ indicates the direction of propagation of the wave. From (4.7), using (4.6) the rotational motions induced by a Love wave are found to be

ωx(p, t) = 0

ωy(p, t) = 0 (4.8) 1 ω (p, t) = ακ sin(ωt κ p + ϕ) . z 2 − ·

Rotational motions induced by a single Love wave are limited to the ωz component. Rotations are scaled of a factor κ/2 with respect to the wave amplitude α. 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 the earth and containing the direction of propagation of the wave. The particle displacement generated by a single Rayleigh wave is u (p, t) = α sin ξ cos ψ cos(ωt κ p + ϕ) x − · u (p, t) = α sin ξ sin ψ cos(ωt κ p + ϕ) (4.9) y − · u (p, t) = α cos ξ cos(ωt κ p + π/2 + ϕ) . z − · The angle ξ is called ellipticity angle of the Rayleigh wave and determines the eccentricity 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∈ oscillation− on the vertical component uz is shifted by +π/2 radians with respect to the oscillation on the direction of propagation). If ξ (0, π/2) the wave is said to be prograde. For ξ = 0 and ξ = π/2 the polarization∈ is vertical and horizontal, respectively. The quantity tan± ξ is known as the ellipticity of the Rayleigh wave. See Maranò et al. (2012)| | for a detailed description of this parametrization. From (4.9), using (4.6) the rotational motions for a Rayleigh wave are found to be

ω (p, t) = ακ sin ψ cos ξ cos(ωt κ p + ϕ) x − · ω (p, t) = ακ cos ψ cos ξ cos(ωt κ p + ϕ) (4.10) y − − · ωz(p, t) = 0 .

Rotational motions induced by a single Rayleigh wave are limited to the ωx and ωy 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 instants by means of instrumentation able to measure the translational motions and the rotational motions. At each location a sensor measures the ground translation along the direction of the axes of the coordinate system x, y, z and the ground rotation 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 surface of the earth positioned at locations pn n=1,...,Ns . We restrict our interest to small aperture arrays and work with{ a flat} earth model. Each signal is sampled at K instants tk k=1,...,K . The recording from the six components of the n-th sensor are grouped{ } in six channels numbered from ℓ = 6n 5 to ℓ + 5 = 6n and (6n−5) (6n−4) −(6n−3) ordered as uk = ux(pn, tk), uk = uy(pn, tk), uk = uz(pn, tk), (6n−2) (6n−1) (6n) uk = ωx(pn, tk), uk = ωy(pn, tk), and uk = ωz(pn, tk). We let L = 6Ns denote the total number of channels in an array of six components sensors. Each measurement is corrupted by independent additive Gaussian noise Z(ℓ) (0, σ2). Noise variance is, in general, different on each channel k ∼ N ℓ

(ℓ) (ℓ) (ℓ) Yk = uk (θ) + Zk . (4.11)

(ℓ) The quantities uk (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

L K (ℓ) (ℓ) 2 1 (uk (θ) yk ) pY (y θ) = exp − , (4.12) 2 2σ2 | 2πσℓ − ℓ ! ℓY=1 kY=1 p (ℓ) where we grouped the measurements as Y = Yk ℓ=1,...L . { }k=1,...,K Whenever a sensor has less than six components, the corresponding missing channels 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 measurements is considered. For a comprehensive introduction to estimation theory, we refer the interested reader to Kay (1993).

System model. We consider a sinusoid with known angular frequency ω. The sinusoid is a deterministic function of amplitude α and phase ϕ, which are unknown. Noisy measurement of the sinusoid are taken at K known time instants t . Each measurement Y is corrupted by independent additive noise { k}k=1,...,K k Zk as

Yk = α cos(ωtk + ϕ) + Zk , for k = 1,...,K. We assume the statistical properties of the noise to be known. Speciﬁcally, the noise has Gaussian distribution with zero mean and known variance σ2, i.e., Z (0, σ2). k ∼ N Probability density function. Given the assumption of independent and iden- tically distributed noise, it is straightforward to write the probability density function (PDF) of the measurements as

K 2 1 (α cos(ωtk + ϕ) yk) pY (y θ) = exp − . | √2πσ2 − 2σ2 kY=1 The parameter vector θ = (α, ϕ) collects the parameters of the model that are unknown and need to be estimated.

Likelihood function. Given measurements y˜ = y˜ , the LF of the { k}k=1,...,K observations is pY (y˜ θ). Observe that the likelihood function (LF) is a function of θ and the measurements| y˜ are ﬁxed. Given the observations y˜, the LF quantiﬁes how likely are the parameters θ.

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

θˆ = argmax pY (y˜ θ) . 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 estimator should be unbiased. On the average we expect the estimator to provide the true value, i.e. E θˆ θ = 0, where E denotes the expected value. Second, the estimator should{ − be} accurate. Or,{·} in other words, the estimator variance E (θˆ θ)2 should be as small as possible. Under certain assumptions, ML estimators{ − } are unbiased and have smaller variance that any other unbiased estimator. 71

An Example of the Estimation of a Sinusoid in Noise (continued).

Fisher information. Fisher information (FI) quantiﬁes the information we obtain about each parameter from our experiment. The Fisher information matrix (FIM) is deﬁned as

2 ∂ ln pY (y θ) I(θ) = E | , − ∂θ2 which can be interpreted as the average Hessian matrix at the point θ of the negative log-likelihood function (LLF). Observe that the FIM depends on the statistical model and on the parameter vector θ but is independent of the measurements. For our model, the FIM is

K 1 0 I θ ( ) = 2 . 2σ 0 α !

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

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

K 2 ′ 1 (α cos(ωtk + ϕ1 + ϕ2) yk) pY y θ = exp − , √2πσ2 − 2σ2 k=1 Y ′ where the parameter vector is θ = (α, ϕ1, ϕ2). It is evident, that there is some ambiguity in this parametrization since there are infinite ϕ1, ϕ2 pairs defining the same sinusoid. Therefore two distinct parameter vectors defining the same PDF exist and thus the model is not identifiable. When such ambiguity is not immediately evident, another way to verify whether a model is identifiable or not, is to test the singularity of the FIM. For this latter model, is found

1 0 0 K I(θ′) = 0 α α , 2σ2   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 (continued).

Cramér-Rao bound. The accuracy of any unbiased estimator is limited by the Cramér-Rao bound (CRB). In other words, for a given statistical model there is no unbiased estimator having variance smaller than the CRB. In practice, the CRB is obtained from the elements on the main diagonal of the matrix inverse of I(θ). In our example, the computation of I−1(θ) is straightforward since the matrix I(θ) is diagonal. From the elements on the main diagonal of I−1(θ), it is found that the variance of amplitude and phase estimates are lower bounded as 2σ2 E (ˆα E αˆ )2 − { } ≥ K 2σ2 E (ϕ ˆ E ϕˆ )2 . − { } ≥ αK This analytic result provides insights useful for the design of the experiment and a benchmark that allows to evaluate the performance of an estimation algorithm.

4.3 Theoretical Performance Analysis

In this section, we discuss the advantages and potential of the joint processing of rotational and translational measurements from a theoretical standpoint. To this aim, we use several ideas from estimation theory. A reader unfamiliar with this branch of statistics may refer to the box “An Example of the Estimation of a Sinusoid in Noise” included in this article. First, we derive an expression of the Fisher information matrix (FIM) for each wave models considered. Then we look at the issue of the identifiability of statistical models concerning Love wave and Rayleigh wave for three components (translational) single sensor, six components (translational and rotational) single sensors, and arrays of sensors. Following, we find the smallest achievable mean-squared estimation error (MSEE) of an unbiased estimator using the Cramér-Rao bound (CRB). The contribution to the parameter estimation of the different measurements and parameters is also understood. At last, we briefly discuss the estimator performance at lower SNR, in the threshold region.

Introduction The MSEE of an estimation algorithm can be computed numerically using Monte Carlo methods. For example, it is sufficient to repeat a large number of times the estimation of the wavefield parameters of a known wave with different noise realizations to compute the MSEE. In this way, it is possible to quantitatively compare the estimation accuracy of two different estimation algorithms or the estimation accuracy of the same estimation algorithm under different conditions. The CRB, provides a lower bound on estimator variance and is independent of estimation technique and algorithm implementation. Therefore 73

10−3

10−4

10−5

10−6 MSEE

10−7 Threshold zone region

10−8 No information Asymptotic region

10−9 −20 −15 −10 −5 0 5 10 SNR [dB] Figure 4.1: An example of the MSEE of a ML estimator. The MSEE is depicted with a blue dashed line. In the no information region the MSEE is very large and constrained by the implementation of the algorithm. In the threshold region the occurrence of outliers keep the MSEE significantly larger than the CRB. At last, in the asymptotic region, the MSEE is well described by the CRB, which is shown with the black dashed line. the MSEE of any algorithm can be compared with the CRB. Fig. 4.1 illustrates these concepts with an example. It is known from literature that non-linear estimators exhibit an abrupt increase in the MSEE below a certain SNR or sample size. This behavior is called threshold effect and is due to a transition from local to global estimation errors (Van Trees, 2001). Three operation regions for the estimator are defined at different SNR ranges, see also Fig. 4.1. At very low SNR, the noise dominates over 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 global estimation errors occur. Global estimation are also known as outliers. At even larger SNR is found the asymptotic region. Local estimation error occurs in this region and the MSEE of a ML estimator is well described by the CRB.

Preliminary deﬁnitions: Consider the following deﬁnitions related to the geometrical layout of the array. We introduce the coordinate system (a, b), which is related to (x, y) as

a cos ψ sin ψ x = , (4.13) b sin ψ cos ψ y ! − ! ! where the angle of rotation is the azimuth ψ. Therefore a is the axis along the direction of propagation of the wave and b the axis perpendicular to it. In this rotated coordinate system we consider the new sensor positions (a , b ) . The moment of inertia (MOI) of the array in the coordi- { n n }n=1,...,Ns 74 nate system (a, b) are deﬁned as

Ns Q = (a a¯)2 (4.14) aa n − n=1 X Ns Q = (b ¯b)2 (4.15) bb n − n=1 X Ns Q = (a a¯)(b ¯b) , (4.16) ab n − n − n=1 X 1 Ns ¯ 1 Ns where a¯ = Ns n=1 an and b = Ns n=1 bn deﬁne the phase center of the array. We observeP that the MOIs are invariantP to a translation of the array and that 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 diﬀerent measurements contribute to the parameter estimation of the statistical model. The Fisher information (FI) conveys the amount of information about a statistical 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

2 ∂ ln pY (y θ) I(θ) = E | , (4.17) − ∂θ2 where E denotes the expectation operation. The vector θ collects the unknown waveﬁeld{·} parameters of either Love wave or Rayleigh wave. The matrix I is a square symmetric matrix with as many columns as the elements in the vector θ. Measurement of translational and rotational motions are independent corrupted by additive white Gaussian noise, as in (4.11). Throughout this section we consider the translational and the rotational components to be subject to 2 2 diﬀerent noise levels, with power σt and σr respectively. For independent observations, the FIM is additive. Let It(θ) and Ir(θ) be the FIM pertaining the translational and the rotational components. The FIM accounting 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 of the derivation, are provided. The FIMs for the model of a single Love wave are 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) and in (4.43) for translational and rotational measurements, respectively. 75

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

Identiﬁability

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

pY (y θ ) = pY (y θ ) θ = θ θ , θ Θ . (4.19) | 1 | 2 ⇔ 1 2 ∀ 1 2 ∈ This definition means that two distinct parameter vectors which specify the same 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 limited to the local identifiability, i.e., to a neighborhood of the maximum likelihood point. In addition, a statistical model is identifiable if and only if the corresponding FIM 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 (translational) sensor. From (4.7), we understand that this model is not identifiable as several parameters specify the same PDF. Consider, for exam- (L) T (L) T ple, the parameter vectors θ1 = (α, ϕ, κ, ψ) , θ2 = (α, ϕ, γκ, ψ) with R (L) T γ +, and θ3 = (α, ϕ, κ, ψ + π) , they specify the same distribution, i.e. ∈ (L) (L) (L) pY (y θ1 ) = pY (y θ2 ) = pY (y θ3 ). Indeed, with a single translational sensor| it is not possible| to determine| the wavevector, and thus the velocity of propagation, of the Love wave. Moreover, there is an ambiguity of 180◦ about the 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 (L) (L) two distinct parameter vectors θ1 = θ2 specifying the same distribution do not exist. This fact can be verified by6 checking the non-singularity of the FIM in (4.38) for Ns = 1. The same conclusion has been reached using different arguments 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 (translational) sensor. From (4.9), we understand that this model is not identifiable as several parameters specify the same distribution. Indeed the parameter vectors θ(R) = (α, ϕ, κ, ψ, ξ)T, θ(R) = (α, ϕ, γκ, ψ, ξ)T with γ R , and 1 2 ∈ + θ(R) = (α, ϕ, κ, ψ + π, ξ)T specify the same PDF. Again, from a single sensor 3 − 76 is not possible to retrieve any information concerning wave velocity of propagation. Moreover, there is an ambiguity involving direction of propagation and the prograde/retrograde sense of rotation. For a six components sensor the estimation of θ(R) is well-posed. This can be understood from (4.9) and (4.10). Again, this can be verified by checking the 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 or a Rayleigh wave. The only exception is the case of collinear sensors. Indeed a linear array cannot resolve the wavenumber for a wave propagating perpendicular to the array. When employing an array of six components (translational and rotational) the limitation of the linear array is no more present. Both these facts can be veriﬁed by testing the singularity of the FIMs. Interestingly, with an array of three components (rotational) sensors it is possible to identify the parameters of a Love wave but not the parameters of a Rayleigh wave. Indeed an array of sole rotational sensors it is not capable of estimating correctly Rayleigh wave amplitude, phase, and ellipticity. The (R) (R) parameter vectors θ1 = (α, ϕ, κ, ψ, ξ), θ2 = (γα, ϕ, κ, ψ, arccos(cos ξ/γ)) with γ R , and θ(R) = (α, ϕ+π, κ, ψ, ξ) specify the same statistical model. ∈ + 3 − Cramér-Rao Bound The Cramér-Rao bound (CRB) is a lower bound on the variance of unbiased estimators (Cramér, 1946; Rao, 1945). Knowledge of a lower bound on the estimator variance has at least two practical implications. First, it allows us to evaluate the performance of an estimation algorithm, by enabling a quantitative comparison between the mean-squared estimation error (MSEE) of the algorithm under test and the smallest achievable variance. Second, the analytic expression of the CRB enables us to design the experiment set up in order to reduce the lower bound and therefore increase the amount of information gathered by the experiment. The information inequality states that the MSEE of an unbiased estimator is lower bounded as

T E θˆ E θˆ θˆ E θˆ (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 provide a lower bound on the MSEEs of the corresponding parameters. In high SNR regime, the CRB well describes the performance of ML estimator. Thus in order to increase estimation accuracy, one is interested to reduce the CRB. This can be achieved by tuning the value of some deterministic parameters of the model as, for example, increasing the number of sensors or optimizing the array geometry. 77

To derive the CRB for the waveﬁeld parameters of interest is necessary to invert the FIM I. The CRB is obtained from the elements on the main diagonal of I−1. Since we are interested in the elements on the main diagonal of I −1 corresponding to wavenumber and ellipticity angle, we avoid the complete inversion of I as follows. We partition the FIM as

c dT I(θ) = , (4.21) d G ! where c is a scalar, d is a vector, and D is a matrix of suitable sizes. The element in the ﬁrst position of I−1 is then found using the Woodbury matrix identity to be

−1 (I(θ)) −1 = c dTG−1d , (4.22) 1,1 − 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 been referred to by some authors− as equivalent Fisher information (EFI) (Shen & Win, 2010). In contrast with FI, the EFI accounts for the uncertainty introduced by the other unknown parameters of the statistical model. The term c is exactly the FI of the parameter of interest. The term dTG−1d is non-negative since G is PSD being a diagonal sub-block of a PSD matrix. This last quantity accounts for the uncertainty due to the other parameters. It is now clear that reducing the CRB is equivalent to increase the EFI. In other words, increasing the EFI is desirable as better estimation accuracy can be achieved. In order to use (4.22) eﬀectively, it may be necessary to permute the row and columns 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′ = PTIP . In the following, we restrict ourselves to the analysis of the CRB of wavenumber and ellipticity angle as these are the parameters of greater practical interest.

Love wave wavenumber The CRB on Love wavenumber for translational measurements is obtained using (4.22) and (4.40). The MSEE of Love wave wavenumber is lower bounded as

α2K Q2 −1 E (ˆκ E κˆ )2 Q ab . (4.23) − { } ≥ 2σ2 aa − Qκ + N / 2 t bb s 2 The CRB is directly proportional to noise power σt , inversely proportional to the amplitude of the wave α and to the number of samples K. We observe that K 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 variability of real seismic sources. We emphasize that the CRB depends on the sensor positions p only trough the MOIs. The term Q is representative { n}n=1,...,Ns aa 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 direction of wave propagation a, cf. (4.14). However observe that a large aperture may invalidate the plane wave assumption. The last term is due to the uncertainty of the other waveﬁeld parameters and increases the CRB. It can be eliminated by choosing an array geometry such that Qab = 0. The CRB on Love wavenumber for rotational measurements is obtained using (4.22) and (4.41). The MSEE of Love wave wavenumber is lower bounded as

α2κ2K Q2 −1 E (ˆκ E κˆ )2 Q ab . (4.24) − { } ≥ 8σ2 aa − Q r bb The CRB is similar to the expression in (4.23). One difference is the presence of a factor 4/κ2. This is due to the different overall amplitude of the signal measured on ωz. Concerning seismic surface waves, the κ is generally a small quantity (smaller than one) thus the CRB is increased. In addition, the smaller the wavenumber, the less information is obtained from this type of measurement. The CRB on Love wavenumber for joint translational and rotational measurements 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 first and the third addends of the sum are, similarly to (4.23) and (4.24), representative of the information contributed by the spatial sampling of the wavefield and the uncertainty due to the other parameters weighted by the quality of the signal on the translational and the rotational components. The second addend is representative of the information gain due to the joint processing of the translational and rotational measurements. This term is proportional to Ns and does not go to zero in the single sensor case, so that a single rotational sensors carries information about the wavenumber.

− 2 2 2 1 CtCrNs/4 Q Ct + κ Cr/4 E (ˆκ − E{κˆ})2 ≥ C + κ2C /4 Q + − ab t r aa 2 2 2 Ct + Crκ /4 CtNs/κ + (Ct + κ Cr/4) ! (4.25) −1 2 2 2 CtCrNs cos ξ Q Φ E (ˆκ − E{κˆ})2 ≥ ΦQ + − ab aa 2 2 2 2 Ct + κ Cr CNt sin ξ s/κ + Cr cos ξNs + ΦQbb ! (4.26)

2 E (ξˆ− E{ξˆ}) ≥ (CtNs − n o 2 2 2 1 κ Φ QaaΨNs − κ (Q − QaaQbb)Φ + ab 2 2 2 2 2 3 NsCrCt cos ξ(ΨNs + κ ΦQbb) + QaaΨΦ Ns − κ (Qab − QaaQbb)Φ ! (4.27)

2 2 2 2 Ct = α K/2σt Φ = Ct + Crκ cos ξ with 2 2 and 2 2 2 Cr = α K/2σr Ψ = Ct sin ξ + Crκ cos ξ . 79

Rayleigh wave wavenumber The CRB on Rayleigh wavenumber for translational measurements is obtained using (4.22) and (4.42). The MSEE of Rayleigh wave wavenumber is lower bounded as

2 2 −1 2 α K Qab E (ˆκ E κˆ ) Qaa . (4.28) − { } ≥ 2σ2 − Q + N sin2 ξ/κ2 t bb s 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 lower bounded as

α2κ2 cos2(ξ)K Q2 −1 E (ˆκ E κˆ )2 Q ab (4.29) − { } ≥ 2σ2 aa − Qκ + N / 2 r bb s This result is similar to (4.24) and the same considerations apply. Observe that when the Rayleigh wave is horizontally polarized (ξ = π/2) the EFI is zero since no rotations are induced by the Rayleigh wave. ± The CRB on Rayleigh wavenumber for joint translational and rotational measurements is obtained using (4.22) with the FIM for the joint measurements (4.39). The MSEE of Rayleigh wave wavenumber is lower bounded as in (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 Rayleigh± wave and thus no information is added by the rotational measurements.

Rayleigh wave ellipticity angle The CRB on Rayleigh ellipticity angle for translational measurements is obtained using (4.22) and (4.42). The MSEE of Rayleigh wave ellipticity angle is lower bounded as

2 −1 E ˆ E ˆ 2 α K (ξ ξ ) 2 Ns . (4.30) − { } ≥ 2σt n o This quantity is related to the number of sensors and it is not aﬀected by the geometry of the array. As discussed earlier in this section, an array of sole rotational sensors is not able to estimate Rayleigh wave ellipticity angle as the model is unidentiﬁable. The CRB on Rayleigh ellipticity angle for joint translational and rotational measurements is obtained using (4.22) with the FIM for the joint measurements (4.39). The MSEE of Rayleigh wave ellipticity angle is lower bounded as 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 of global errors are well known in literature (Hawkes & Nehorai, 1998; Cox & 80

Lai, 2007). Performance in the threshold zone of direction of arrival estimators is studied in detail in Athley (2008). In this work, this issue is not addressed directly, however we emphasize that the use of additional measurements reduces the magnitude of local maxima other than the true maximum of the LF and thus of global errors. Improvement in accuracy are to be expected in the low SNR regime, i.e. threshold zone. Performance in the threshold zone are not easily quantiﬁable analytically and we limit ourselves in presenting some numerical 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 parameters for Love waves and Rayleigh waves relying on observation from seismic sensors. The method models jointly measurements from all sensor components making optimal use of the available information. The wavefield parameters are also estimated jointly. The noise variance on each channel is estimated adaptively. Information from the different channel is merged according to the different noise levels on the different sensor components. In our approach, we model the system by means of a probabilistic graphical model. A complex system where a large number of random variables and statistical parameters interact with complex relationships can be effectively represented by a graphical model. Within the graphical model, observed random variables (measurements), unobserved random variables, and parameters of the statistical model are represented in a unique framework together with the functional relationships occurring among them. The probabilistic graph can be used to perform inference tasks in an efficient manner. As an example, likelihood of the observations and thus ML estimation can be performed exploiting the structure of the graph. By using the graph it is possible to understand the relationship between the different parts of the stochastic system and then, for example, derive sufficient statistics which enable to efficiently compute statistical quantities of interest. In our approach we rely on factor graphs, one flavor among many graphical modeling techniques (Kschischang et 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 the parameter space allows us to obtain a ML estimate θˆ

θˆ = argmax pY (y˜ θ) . (4.31) θ | In this context, diﬀerent sensor technologies are used and the amplitudes of the measured signals are expected to vary greatly. It would be surely not optimal to assume equal noise variance on every channel. Thus, after estimating the waveﬁeld parameters the noise variances are also estimated as

2 2 ˆ 2 2 (ˆσ1 ,..., σˆL) = argmax pY (y˜ θ, σ1, . . . , σL) , (4.32) 2 2 | (σ1 ,...,σL) where θˆ is the estimated waveﬁeld parameter vector obtained from (4.31). 81

The maximizations in (4.31) and in (4.32) are repeated alternatively and the estimation of the wavefield parameters accounts for the different noise level on the different sensors. Since the likelihood is a finite value the alternating maximizations are guaranteed to converge. In our implementation, the maximizations in (4.31) and in (4.32) are not computed directly from (4.12). Details concerning the functioning and the implementation of our algorithm are found in Maranò et al. (2012) and references therein. 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 derive a sufficient statistic. This allows to perform ML parameter estimation in a computationally 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 no preprocessing to the recordings other than mean removal. The whole signal is split in time windows where the signal is assumed to be stationary. The length of such time windows is non-adaptive and not dependent on frequency. We deﬁne the SNR as

α2 SNR = 2 (4.33) 2σt 2 where σt is the noise variance on the translational components. It is clear from equations (4.8) and (4.10) that rotational motions have significantly smaller amplitude than translational motions. Depending on the value of the wavenumber κ, rotational motions can be even one or two order of magnitude smaller than the translational counterparts. In the numerical examples that follow, we choose the true value of the 2 2 2 variance on the rotational components to be σr = κ σt . This choice is motivated by the fact that the different noise level allows to obtain measurements of comparable SNR on translational and rotational components. Both in the synthetic examples and in the real dataset the noise variances are unknown to the 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 potential and the benefit introduced by the joint processing of translational and rotational components over the processing of the sole translational components. In Sec. 4.5 we quantify increased estimation accuracy, in terms of MSEE, achieved by employing the rotational measurements and compare with the CRB. At last in Sec. 4.5 and 4.5 we show two applications from translational and rotational recordings 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 translational components and rotational components enables us to identify statistical models for surface waves and to estimate correctly the wave parameters. 82

0.1 0.1 [1/m] [1/m] y y

0 0 Wavenumber along Wavenumber along

−0.1 −0.1 −0.1 0 0.1 −0.1 0 0.1 Wavenumber along x [1/m] Wavenumber along x [1/m] (a) A line across the origin individuate the (b) The LLF exhibits a single maxima. maxima point of the LLF reﬂecting the inability to determine velocity of propagation and direction of propagation.

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

The figures shown in this section and in the following should be seen as explanatory examples. In first place, a different noise realization will lead to 2 2 a different LF . More importantly, a different choice of σt and σr could lead to a substantially different shape of the LF. To reduce this effects and to 2 2 2 ensure a fair comparison, we use a high SNR (SNR = 10 dB) and σr = κ σt . We consider 1 s of observation, sampling at 100 Hz monochromatic waves of frequency 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 single Love wave with θ(L) = (1, 0, 0.05, π/4)T is considered. Fig. 4.2(a) depicts the LLF obtained from a single three components (translational) sensor. A whole set 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. This reflects≥ the∈ inability { to determine}} wavenumber and azimuth from a single three components sensor. Fig. 4.2(b) depicts the LLF obtained from a single six components (translational and rotational) sensor. The global maximum point is seen in correspondence of the true wavefield parameters. Indeed a single six components sensor allows the determination of velocity of propagation and direction of propagation without ambiguity. A similar setup is repeated for a single Rayleigh wave with parameters θ(R) = (1, 0, 0.05, π/4, π/4)T . In Fig. 4.3 the LLFs of observations of a noisy Rayleigh wave are− shown. We are interested in showing the shape of the LLF as a function of three parameters, namely wavenumber, ellipticity, and azimuth. Thus we depict three slices of the LLFs, each slice is a function of two 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, namely the 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∨ reflects the inability to} determine unambiguously wavenumber, azimuth, and sense of rotation 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 components (translational and rotational) sensor. The global maximum point is seen in correspondence of the true wavefield parameters. Indeed a single six components sensor allows the determination of velocity of propagation, direction of propagation, and Rayleigh wave ellipticity without ambiguity. From the previous pictures we can empirically confirm the theoretical findings about model identifiability of Sec. 4.3 and that, at least under the good conditions of high SNR, the joint processing of translational motions and rotational motions allows to estimate all the wavefield parameters correctly. The broadness of the main peak of the LF suggests that the κ remains difficult to estimate accurately with a single six components sensor. This latter aspect is quantified 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 components sensors and six components sensors. We consider an array of five sensors arranged on a circle of radius 20 m as shown in Fig. 4.4. The same choice of wavefield parameters and SNR of Sec. 4.5 is used in this section. In Fig. 4.5 the LLFs of observations of a single Love wave are shown. It is shown that the local maxima (sidelobes) of the LLF are smaller in the six components case than in the three components case. In Fig. 4.6 different slices of the LLFs of observations of a single Rayleigh wave 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) with Fig. 4.6(d). By comparing the LLFs in this Section with the corresponding LLFs of Sec. 4.5, we see that using the five sensors array the wavenumber and the azimuth are more easily determined, as witnessed by the peakiness of the LLFs in the two different setups. The reason is that estimation from an array of sensors relies on the very important information extracted from the spatial sampling 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 components and six components with the theoretical bounds given by the CRB derived in Sec. 4.3. Fig. 4.7 portrays the MSEE obtained by means of Monte-Carlo simulations with diﬀerent processing settings as a function of SNR. The ﬁve sensors array depicted in Fig. 4.4 is considered, with three components sensors and with six 84

0.1 0.1 [1/m] [1/m] y y

0 0 Wavenumber along Wavenumber along

−0.1 −0.1 −0.1 0 0.1 −0.1 0 0.1 Wavenumber along x [1/m] Wavenumber along x [1/m] (a) Slice of the LLF for ξ = −π/4 (retrograde (b) Slice of the LLF for ξ = −π/4. The LLF motion). exhibit a single maxima.

π π 2 2

[rad] π [rad] π

ξ 4 ξ 4

0 0

- π - π

Ellipticity angle 4 Ellipticity angle 4

- π - π 2 0 π/2 π 3π/2 2π 2 0 π/2 π 3π/2 2π Azimuth ψ [rad] Azimuth ψ [rad] (c) Slice of the LLF as a function of ξ and ψ (d) Slice of the LLF as a function of ξ and ψ for κ = 0.05. The likelihood is maximized for for κ = 0.05. Direction of propagation and opposite direction of propagation with diﬀer- sense of rotation are pinpointed correctly. ent sense of rotation, prograde (π/2) or retrograde (−π/2).

π π 2 2 [rad] [rad] π π 4 ξ

ξ 4

0 0

- π - π

Ellipticity angle 4 Ellipticity angle 4

- π - π 2 0 0.02 0.04 0.06 0.08 0.1 2 0 0.02 0.04 0.06 0.08 0.1 Wavenumber [1/m] Wavenumber [1/m] (e) Slice of the LLF as a function of κ and ξ (f) Slice of the LLF as a function of κ and ξ for ψ = π/4. The function is constant for dif- for ψ = π/4. The wavenumber can be cor- ferent wavenumbers, because any wavenum- rectly estimated using a six components sen- ber value ﬁts the data equally well. sor.

Figure 4.3: LLFs of observations from a single sensor of a single Rayleigh wave as a function of wavenumber κ, azimuth ψ, and ellipticity angle ξ. Comparison of analysis of sole translational components (left) and joint translational and rotational components (right). Large LL values are shown with colors towards red and low LL values with colors towards blue. White crosses and lines mark the maxima point. 85

0 y [m] −5

−10

−15

−20 −15 −10 −5 0 5 10 15 20 x [m]

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

0.1 0.1 [1/m] [1/m] y y

0 0 Wavenumber along Wavenumber along

−0.1 −0.1 −0.1 0 0.1 −0.1 0 0.1 Wavenumber along x [1/m] Wavenumber along x [1/m] (a) LLF obtained from translational compo- (b) LLF obtained from translational and rota- nents only. tional components jointly.

Figure 4.5: LLFs of observations from a five sensors array of a single Love wave as a function of wavenumber κ and azimuth ψ. Comparison of analysis of sole translational components (left) and joint translational and rotational components (right). Large LL values are shown with colors towards red and low LL values with colors towards blue. White crosses mark the maxima point. components sensors. We also consider the performances of a single six components sensor. The wavefield parameters are the same used in the numerical 2 examples of the previous section and are unknown to the algorithm. Both σt 2 2 and σr are unknown to the algorithm and estimated. The true values σt and 2 σr are chosen as explained in the introduction of this section in order to have comparable SNR on both sensor types. In Fig. 4.7(a) the estimation of Love wave wavenumber is analyzed. At very low SNR, where the noise dominates, the estimate is substantially random. The MSEE saturates for decreasing SNR since the wavenumber estimate is constrained by the algorithm implementation to belong to a finite interval. As the SNR increases, the ML method using six components always exhibits the 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 0.1 [1/m] [1/m] y y

0 0 Wavenumber along Wavenumber along

−0.1 −0.1 −0.1 0 0.1 −0.1 0 0.1 Wavenumber along x [1/m] Wavenumber along x [1/m] (a) Slice of the LLF as a function for ξ = −π/4. (b) Slice of the LLF for ξ = −π/4.

π π 2 2

[rad] π [rad] π

ξ 4 ξ 4

0 0

- π - π

Ellipticity angle 4 Ellipticity angle 4

- π - π 2 0 π/2 π 3π/2 2π 2 0 π/2 π 3π/2 2π Azimuth ψ [rad] Azimuth ψ [rad] (c) Slice of the LLF as a function of ξ and ψ (d) Slice of the LLF as a function of ξ and ψ for κ = 0.05. for κ = 0.05.

π π 2 2

[rad] π [rad] π 4 ξ ξ 4

0 0

- π - π

Ellipticity angle 4 Ellipticity angle 4

Figure 4.6: LLFs of observations from a five sensor array of a single Rayleigh wave as a function of wavenumber κ, azimuth ψ, and ellipticity angle ξ. Com- parison of analysis of sole translational components (left) and joint translational and rotational components (right). Large LL values are shown with colors towards red and low LL values with colors towards blue. White crosses mark the maxima point. 87 six components array over the three components array is substantial. This aspect was discussed in Sec. 4.3. For sufficiently large SNR, the ML method using six components achieves the CRB. Even for high SNR the three components array does not achieve the CRB as it disregards the rotational measurements. It is worth noting that the performance gap observed when comparing the three components array with the six components array in the asymptotic re- 2 2 2 gion is strongly dependent on the choice of σt and σr . For very large σr , the rotational measurements become uninformative and the performance gap will 2 narrows to zero. On the other hand, for very large σt the translational measurement become uninformative and therefore the three components array will exhibit poor performance while the six components will still provide meaningful estimates.The performance gap between single sensor and any of the two five sensors arrays is considerably large. In order to achieve with a single six components sensor the same MSEE achieved with an array of sensors it is would be necessary a SNR of several decibels higher. This gap is explained by the fact that the single sensor only relies on amplitude information and disregards the phase information relative to wave propagation. This is also quantified analytically by the expression of the CRBs in Section 4.3. In Fig. 4.7(b) the estimation of Rayleigh wave wavenumber is analyzed. In this 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 smaller MSEE than the three components array over a certain SNR range.Concerning the Rayleigh ellipticity angle estimation, the performance gap is smaller than the previous case. Indeed the CRB on Rayleigh wave ellipticity for a translational 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 demolition in southern Germany (Wassermann et al., 2009). The seismic motion is recorded 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 with a translational sensor at the central location labeled ’BW01’. The translational velocimeters 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 is split in 0.75 seconds window which are processed independently with a 50% overlap. We apply the considered ML method and process the signals recorded by the 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 darker colors corresponding to a value being more often estimated. The red dashed lines represent a manual pick of upper and lower boundaries of the dispersion curve as identiﬁed by visual inspection. The red dots depict the estimated dispersion curve starting from such manual selection obtained as the median of the values in the selection. The blue lines represent constant velocity lines. 88

10−3

10−4

−5

] 10 2 −

10−6

−7

MSEE [m 10 3C ML (Ns = 5)

6C ML (Ns = 5) 10−8 6C ML (Ns = 1) Cramér-Rao L.B. 10−9 −25 −20 −15 −10 −5 0 5 10 15 20 SNR [dB] (a) Love wavenumber MSEE.

10−3

10−4

−5

] 10 2 −

10−6

−7

MSEE [m 10 3C ML (Ns = 5)

6C ML (Ns = 5) 10−8 6C ML (Ns = 1) Cramér-Rao L.B. 10−9 −25 −20 −15 −10 −5 0 5 10 15 20 SNR [dB] (b) Rayleigh wavenumber MSEE.

100

10−1 ] 2 10−2

10−3

MSEE [rad 3C ML (Ns = 5)

6C ML (Ns = 5) 10−4 6C ML (Ns = 1) Cramér-Rao L.B.

−25 −20 −15 −10 −5 0 5 10 15 20 SNR [dB] (c) Rayleigh ellipticity angle MSEE.

Figure 4.7: Comparison of the MSEE from different processing setups with the CRB at different SNR. The processing of three components (3C) translational sensors is compared with six components (6C) translational and rotational sensors. Different number of sensors Ns is also compared. 89

BW05

20 BW08 BW06 BW03

0 BW01 y [m] BW07 −20

−40 BW02

−40 −20 0 20 40 x [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 rotational sensors. The array is centered in 48.108589◦ N 11.582967◦ E.

Empirical array resolution limits according to Asten & Henstridge (1984) are depicted with thin dashed black lines. The stripes visible in the ﬁgure are due to the fact that the wavenumber estimates within the same time window and at neighboring frequencies are strongly correlated. In other words, the maximum of the LF changes only slightly when the frequency ω is changed slightly. The ellipticity angle is shown in Fig. 4.9(b). Rayleigh particle motion is retrograde, i.e. ξ ( π/2, 0), for frequencies above 4 Hz. Below this frequency the wave appears∈ to− be close to horizontally polarized (ξ = π/2). It should be also considered that at low frequencies the wavenumber estimates± are below the conventional resolution limit. For the single sensor setting, we process the recording of the co-located translational sensor and rotational sensor at position BW01. The wavenumber estimates are shown in Fig. 4.9(c). From this picture is possible to recognize a general increase in the values of the estimated wavenumbers. Compared to the results obtained from the array (Fig. 4.9(a)), it is possible to see a shift of the estimated wavenumbers toward lower values (faster velocities). One possible explanation 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 by the possible presence of higher modes of propagation, as suggested in Kurrle et al. (2010). In the single sensor setting, the dispersion curve does not appear to 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 good agreement 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 some extent larger in the single sensor setting than in the array setting. 90

π 0.025 2 s 0.02 π [rad]

400m / 4 ξ s

0.015 500m /

s 0

0.01 700m / s - π 4 / 1200m Ellipticity angle

Wavenumber [1/m] 0.005

0 - π 1 2 3 4 5 6 7 8 2 1 2 3 4 5 6 7 8 Frequency [Hz] Frequency [Hz] (a) Rayleigh wavenumber estimated using (b) Rayleigh ellipticity angle estimated using the array of translational sensors. the array of translational sensors.

π 0.025 2 s 0.02 π [rad]

400m / 4 ξ s

0.015 500m /

s 0

0.01 700m / s - π 4 / 1200m Ellipticity angle

Wavenumber [1/m] 0.005

0 - π 1 2 3 4 5 6 7 8 2 1 2 3 4 5 6 7 8 Frequency [Hz] Frequency [Hz] (c) Rayleigh wavenumber estimated using a (d) Rayleigh ellipticity angle estimated using single six components sensor. a single six components sensor.

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

The same recording is processed modeling the presence of a single Love wave. In Fig. 4.10(a) wavenumber estimates obtained from the processing of the seven translational sensors are shown. In Fig. 4.10(b) wavenumber estimates obtained from the processing of the sensors co-located at BW01 are shown. Similarly to the case for Rayleigh wave, a shift towards faster velocities is observed. Love wave wavenumber estimates are more scattered than Rayleigh wave wavenumber estimates and the dispersion curve is more diﬃcult to observe. An explanation could be that the Love waves are not as strong because of the nature of the source. Indeed an explosion excites mostly compressional waves and only to a lesser extent Shear waves. The branching of the dispersion curve that can be observed below 4 Hz in Fig. 4.10(a) could also be explained by the little energy of the Love wave.

Analysis of the TAIGER Dataset The TAIGER dataset includes recordings of two explosions in north-eastern Taiwan (Lin et al., 2009). Recordings from an array of eleven accelerometers 91

0.025 0.025 s s 0.02 0.02 400m / 400m / s s

0.015 0.015 500m / 500m / s s

0.01 700m / 0.01 700m / s s / 1200m / 1200m

Wavenumber [1/m] 0.005 Wavenumber [1/m] 0.005

0 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Frequency [Hz] Frequency [Hz] (a) Love wavenumber estimated using the ar- (b) Love wavenumber estimated using a sin- ray of translational sensors. gle six components sensor.

Figure 4.10: Analysis of Love waves for the Agfa dataset. On the left processing from 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 inner locations. In this work, we use recording from the explosion N3P, as named in the 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 is to convert the velocity recordings from the rotational sensors to acceleration (rad/s2). Fig. 4.11(b) shows the wavenumber estimates obtained processing the signals from the translational sensors. It is difficult to determine the wavenumber change with frequency. Associated ellipticity angle estimates are very scattered and not shown. In Fig. 4.11(c) all the available sensors, both translational and rotational are processed jointly. Here the dispersion curve is well identified and the improvement on the previous picture is significant. At last, Fig. 4.11(d) shows the estimated ellipticity angle obtained from the processing of all the sensors. Despite the scatter it is possible to identify a retrograde particle motion at frequencies above 10 Hz. Below this frequency the wave is substantially horizontally polarized. The estimation of the wavenumber and ellipticity angle is significantly improved by the joint processing of translational and rotational sensors over the processing of the sole translational sensors. We observe that the array is rather small, 20 m in diameter, and this might be a limiting factor in the estimation of the wavenumber of seismic waves with wavelengths between 10 m and 50 m. We speculate that in this setting the amplitude information provided by the rotational 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 values, i.e. slower velocity with increasing time. This shift could be explained by the nonlinear behaviors induced by the large strains. This observation however remains speculative, due to missing independent observation such as 92

10 N02A N02B s

0.12 5 N03 200m /

0.1 s 0 N04 N05 N06 N07 N08 0.08 300m / y [m]

0.06 s −5 N09

0.04 500m / s Wavenumber [1/m]

0.02 900m / −10 N10A N10B

0 −10 −5 0 5 10 5 10 15 20 25 x [m] Frequency [Hz] (a) Array layout for the TAIGER dataset. (b) Rayleigh wavenumber estimated using Five rotational sensors are co-located with the array of translational sensors. translational sensors in the inner ﬁve locations of the array. The array is centered at 24.5792222◦ N 121.4818722◦ E.

s π 0.12 2 200m /

0.1 π [rad]

s 4 ξ 0.08 300m /

0.06 0 s

0.04 500m / s - π 4 Wavenumber [1/m] Ellipticity angle

0.02 900m /

0 - π 5 10 15 20 25 2 5 10 15 20 25 Frequency [Hz] Frequency [Hz] (c) Rayleigh wavenumber estimated from the (d) Rayleigh ellipticity angle estimated from joint processing of translational and rota- joint processing of translational and rotational sensors. 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 motions and rotational motions for surface waves. Using tools from statistics, we investigate the contribution of the different measurements types to the accuracy of wavefield parameters estimation. Advantages and limitations of single and array of sensors are outlined quantitatively with respect to identifiability of the statistical models and lower bounds on the estimation accuracy. These findings are also useful for experiment design and to compare estimation algorithms with an implementation-independent benchmark. We show several numerical examples clarifying the theoretical aspects for both single sensor and array settings. 93

A method for ML estimation of wavefield parameters is considered. The method extends a previous work of the authors and accounts for all the measurements and all the wavefield parameters within a single statistical model. In this context we show the estimation of wave parameters of both Love waves and Rayleigh waves from arrays of sensors and single sensors using jointly translational and rotational recordings. In addition, the method accounts for different noise 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 significant when using a single six components sensor instead of a five sensors array. This is due to the lack of information extracted from spatial sampling of the signal and is also explained by our theoretical findings. Secondly we demonstrate, on real recordings, the applicability of the proposed method for the estimation of Love wave and Rayleigh wave parameters for both single sensor and array settings. In the Agfa dataset, we retrieve Love and Rayleigh dispersions curves from single six component sensor and compare with the dispersion curve retrieved from an array of translational sensors. Using a single six component sensor, we observe a shift of the estimated wavenumbers towards faster velocities when compared to the array retrieved dispersion. For the same dataset we also retrieve Rayleigh wave ellipticity angle from the single sensor and find agreement with the same quantity estimated from the array of sensors. Concerning the TAIGER dataset, we compare Rayleigh wave dispersion curve obtained from a three component (translational) array and an array of mixed three- and six-components sensors. We find that the joint analysis of translational and rotational sensors greatly improve the retrieved dispersion curve. It is expected that the interest of the seismological community in this area to grow further in the coming years. As sensor technology will develop further improving the quality and the availability of rotational measurements a wide range 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 the manuscript. The authors also would like to thank the anonymous reviewers for their comments and helping to improve the manuscript. This work is supported in part by the Swiss Commission for Technology and Innovation under project 9260.1 PFIW-IW and with funds of the Swiss Seismological Service.

4.A Derivation of Fisher Information Matrices

The FIM can be derived using (4.17) and (4.12) together with one of the wave model presented in Sec. 4.2. Six distinct FIMs are presented in this section, corresponding to the wave type Love or Rayleigh and the measurement type translational, rotational, or both. 94

The noise variances σℓ ℓ=1,...,L are in general also unknown parameters of the statistical model. However,{ } for the purposes of this discussion, we assume the noise variances to be known and derive the FIM only for the waveﬁeld parameters θ. This choice is supported by the fact that

2 y θ 2 2 E ∂ ln pY , σ1, . . . , σL θ ,σ2 = = 0 , i ℓ ∂θ ∂σ2 I (− i ℓ ) implying that the waveﬁeld parameters and the noise variances are decoupled. Thus, all the derivation to follow can accommodate for the case of unknown noise variances σℓ ℓ=1,...,L with trivial modiﬁcations. With the measurement{ } model presented in (4.12) and assuming the variance 2 on channel ℓ to be known and equal to σℓ , then (4.17) reduces to a simpler expression (Kay, 1993). The element in position i,j of I is obtained as

L K (ℓ) (ℓ) I 1 ∂uk ∂uk [ (θ)]i,j = 2 . (4.34) σℓ ∂θi ∂θj Xℓ=1 Xk=1 2 We further assume that the noise variances are equal to σt for translational 2 measurement and are equal to σr 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 cos2(ωk + β) (4.35) ≈ 2 kX=1 K K sin2(ωk + β) (4.36) ≈ 2 Xk=1 K sin(ωk + β) cos(ωk + β) 0, (4.37) ≈ Xk=1 2πm which are valid for ω being not near 0 or 1/2 and are exact when ω = K , 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 single∈ Love 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 Love wave and rotational measurements, which is given in (4.41). The FIM for the model of a single Love wave using both translational and rotational measurements is obtained adding the two FIM of (4.40) and (4.41) as in (4.18)

(L) (L) (L) I(θ ) = It(θ ) + Ir(θ ) . (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 Rayleigh wave 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 single Rayleigh wave and rotational measurements, which is given in (4.43). The FIM for the model of a single Rayleigh wave using both translational and rotational measurements is obtained adding the two FIM of (4.42) and (4.43) as in (4.18)

(R) (R) (R) I(θ ) = It(θ ) + Ir(θ ) . (4.39)

Ns 0 0 0  α2  Ns ∂Φn Ns ∂Φn 2 0 Ns n=1 ∂κ n=1 ∂ψ (L) α K  P P  It(θ ) =  2  (4.40) 2σ2  0 Ns ∂Φn Ns ∂Φn Ns ∂Φn ∂Φn  t  n=1 ∂κ n=1 ∂κ n=1 ∂ψ ∂κ   P P P  Ns ∂Φ Ns ∂Φ ∂Φ Ns ∂Φ 2  0 n n n Ns + n   Pn=1 ∂ψ Pn=1 ∂ψ ∂κ Pn=1 ∂ψ  Ns 0 Ns 0  α2 ακ  Ns ∂Φn Ns ∂Φn 2 2 0 Ns n=1 ∂κ n=1 ∂ψ (L) α κ K  P P  Ir(θ ) = 2 (4.41) 2  Ns Ns ∂Φn Ns Ns ∂Φn Ns ∂Φn ∂Φn  8σr  +   ακ Pn=1 ∂κ κ2 Pn=1 ∂κ Pn=1 ∂ψ ∂κ   2   0 Ns ∂Φn Ns ∂Φn ∂Φn Ns ∂Φn   Pn=1 ∂ψ Pn=1 ∂ψ ∂κ Pn=1 ∂ψ  2 (R) α K It θ ( ) = 2 · 2σt Ns 0 0 0 0  α2  Ns ∂Φ Ns ∂Φ 0 Ns n n 0  Pn=1 ∂κ Pn=1 ∂ψ   2  ·  0 Ns ∂Φn Ns ∂Φn Ns ∂Φn ∂Φn 0   Pn=1 ∂κ Pn=1 ∂κ Pn=1 ∂κ ∂ψ   2   Ns ∂Φn Ns ∂Φn ∂Φn 2 Ns ∂Φn   0 n=1 ∂ψ n=1 ∂κ ∂ψ Ns sin ξ + n=1 ∂ψ 0   P P P   0 0 0 0 Ns  (4.42) 2 2 2 (R) α κ cos ξK Ir(θ ) = · 2 2σr Ns 0 Ns 0 − tan ξNs  α2 ακ α  Ns ∂Φ Ns ∂Φ 0 Ns n n 0  Pn=1 ∂κ Pn=1 ∂ψ   2  ·  Ns Ns ∂Φn Ns + Ns ∂Φn Ns ∂Φn ∂Φn − tan ξNs   ακ Pn=1 ∂κ κ2 Pn=1 ∂κ Pn=1 ∂ψ ∂κ κ   2   Ns ∂Φn Ns ∂Φn ∂Φn Ns ∂Φn   0 n=1 ∂ψ n=1 ∂ψ ∂κ Ns + n=1 ∂ψ 0   P P P   tan ξNs tan ξNs 2  − α 0 − κ 0 tan 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 seismometers on the surface of the earth. The goal of such measurement surveys is, usually, to estimate the velocity of propagation and the direction of arrival of the seismic waves. In this paper, we address the issue of sensor placement for the analysis of seismic surface waves from ambient vibration wavefields. First, we explain in detail how the array geometry affects the MSEE of parameters of interest, such as the velocity and direction of propagation, both at low and high SNRs. Second, we propose a cost function suitable for the design of the array geometry with particular focus on the estimation of the wavenumber of both Love and Rayleigh waves. Third, we present and compare several computational approaches to minimize the proposed cost function. Numerical experiments verify the effectiveness of our cost function and resulting array geometry designs, leading to greatly improved estimation performance in comparison to arbitrary array geometries, both at low and high SNR levels.

5.1 Introduction

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

96 97 geometry of the sensor array has a major impact on the performance of the array processing system, the design of optimal array geometries is an important task 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 array recordings. Ambient vibrations span a broad range of frequencies and may have natural or anthropic origin (Bonnefoy-Claudet et al., 2006b). Properties of the wavefield, such as the velocity of propagation and polarization, are used to infer a structural model for the site. This has application in microzonation and in geotechnical investigations (Tokimatsu, 1997; Okada, 2006). The wavefield of ambient vibrations is primarily composed of Love waves and Rayleigh waves. Array recordings of ambient vibrations are used to estimate the dispersion curve, i.e. the relationship between the velocity and frequency, of such waves. Fig. 5.1(a) shows the location and the geometry of an array deployed by the Swiss Seismological Service near Brigerbad, in southwestern Switzerland. In this survey, the ground displacement produced by ambient vibrations is recorded for around two hours. A maximum likelihood (ML) method is used to estimate the wavenumbers of Love and Rayleigh waves. Additional details concerning 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 of Rayleigh waves at different frequencies. Darker regions indicate the presence of several wavenumber estimates having the same value. The dark curve extending across the whole figure, from bottom-left to top-right, identifies the dispersion curve of the fundamental mode. The first higher mode is also visible between 8 Hz and 12 Hz, just below the fundamental mode. An ML estimator suffers from two distinct types of error, namely, gross errors (or outliers) and fine errors (Vertatschitsch & Haykin, 1991; Athley, 2008). Both types of errors are influenced by array geometry. At low signal-to-noise ratio (SNR), the presence of local maxima in the likelihood function (LF) leads to large estimation errors. At high SNR, errors are smaller and the variance of the 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 larger wavenumber, there is a significant amount of wavenumber estimates that do not belong to the dispersion curve. These are the gross errors or outliers. On the 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 several different array geometries have been used. In Horike (1985) L-shaped and cross- shaped arrays with a regular sensor spacing have been employed. Irregularly spaced crosses were used in Asten & Henstridge (1984); Milana et al. (1996); Ohori et al. (2002) and Rost & Thomas (2002). In other works, sensors were arranged as several triangles centred around a common point (Satoh et al., 2001b,a). In Gaffet et al. (1998) and Cornou et al. (2003b), concentric circles were used. The limitations of different array geometries have been investigated by different authors. In particular, the interest has been to identify a range of wavenumbers, or a related quantity such as velocity or slowness, where the result of the array processing is more reliable. The largest and the smallest 98

0.1

0.05 Wavenumber [1/m]

0 2 4 6 8 10 12 14 Frequency [Hz] (a) Geometry of the sensor array. The in- (b) Rayleigh wave dispersion curve. Fun- let pinpoints the location of the array within damental mode and ﬁrst higher mode are Switzerland. The geographic coordinates are visible. Swiss coordinates (CH1903).

Figure 5.1: Array deployment and Rayleigh wave dispersion curve from the Brigerbad survey. resolvable wavenumbers have been related either to the array aperture and the smallest interstation distance or to the height of the sidelobes of the array response 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 has also been investigated by the community. Qualitative guidelines, based on empirical evidences, for array design are provided in Rost & Thomas (2002) and Kind et al. (2005). In this work, we present quantitative criteria and computational procedures for designing array geometries for measuring ambient vibrations. Our goal is to improve the performance of the ML estimator of wavefield parameters by optimizing the sensor positions. Given the nature of ambient vibrations and that we mostly rely on very noisy measurements, we are mainly interested in the low SNR regime, focusing primarily on reducing the occurrence of gross errors. In addition, we deal with small scale arrays deployed at the earth surface, and thus optimize the geometry of a planar array, given a budget on the number of 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 gross • errors and fine errors in parameter estimations. We show how the shape of the average LF is related to sensor positions through the Fourier transform of the sampling pattern. We propose a quantitative design criterion for improving estimation per- • formance by means of sensor placement. By reformulating and relaxing the proposed optimization problem, we propose a practical array design algorithm based on a mixed integer program (MIP) with linear objective function and linear constraints. The proposed sensor placement algorithm generates arrays composed of simple regular geometries and are thus suitable for field deployment. 99

We compare the proposed sensor placement techniques with several other • optimization techniques both in terms of array design and of estimation performance. We show through numerical experiments how the estimation performance can be increased by using the proposed array design criterion 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 distinction between gross and fine errors, together with a rigorous derivation of the relationship 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 and propose a design criterion. In Section 5.5, we consider array design methods and in Section 5.6 we compare the results of different techniques. The findings of 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 surveying, a typical goal is to estimate the velocity of propagation and the direction of 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 parameter estimation. Notation. The vector θ indicates a generic value of the wavefield parameters. The vector θ˘ indicates the true, and possibly unknown, wavefield parameters. The vector θˆ indicates an estimate of θ˘. When necessary, a superscript will specify whether the parameter vector describes a Love wave or a Rayleigh wave, i.e., θ(L) and θ(R), respectively. The same convention is used for each element of the wavefield parameter vector.

Seismic Surface Waves The perturbation induced on a ground particle by a seismic wave is described by a vector quantity. Indeed, at each spatial and temporal location it is possible to use, for example, a three component displacement vector to describe the movement of a ground particle in space. In addition, other quantities are available to describe the seismic wavefield, including, for example, ground rotations and strains (Aki & Richards, 1980). Different instrumentation can be used to measure the seismic wavefield and dictates which equations are the more appropriate to model the ground motion induced by the wavefield. Certain instruments can measure the sole vertical displacement of the seismic wavefield, in which case the scalar wave model is suitable. A common type of instrument, the triaxial seismometer, measures the 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 Love wave with a planar wavefront. At the surface of the earth, Love waves exhibit a translational particle motion confined to the horizontal plane, and the particle 100 oscillates perpendicular to the direction of propagation. The particle displacement u = (ux, uy, uz) generated by a single monochromatic Love wave with frequency ω at position p R2 and time t is ∈ u (p, t) = α sin ψ cos(ωt κ p + ϕ) x − − · u (p, t) = α cos ψ cos(ωt κ p + ϕ) (5.1) y − · uz(p, t) = 0 ,

T T where the wavevector κ = κ(cos ψ, sin ψ) = (κx, κy) is a vector pointing in the direction of propagation. The magnitude of the wavevector is called wavenumber κ = κ . The velocity of propagation is ω/κ. The azimuth ψ, measured counterclockwisek k 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 waveﬁeld∈ parameter vector θ(L) = (α, ϕ, κ, ψ). Rayleigh waves exhibit a translational particle motion having an elliptical pattern and conﬁned to the vertical plane perpendicular to the surface of the earth and containing the direction of propagation of the wave. The particle displacement generated by a single Rayleigh wave is

u (p, t) = α sin ξ cos ψ cos(ωt κ p + ϕ) x − · u (p, t) = α sin ξ sin ψ cos(ωt κ p + ϕ) (5.2) y − · u (p, t) = α cos ξ cos(ωt κ p + π/2 + ϕ) . z − · The angle ξ [ π/2, π/2) is called ellipticity angle of the Rayleigh wave and determines the∈ eccentricity− and the sense of rotation of the particle motion. The quantity tan ξ is known as the ellipticity of the Rayleigh wave. See Maranò et 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 parts of this paper to introduce certain ideas before extending them to Love wave and Rayleigh wave models. This wave model is analogous to an acoustic wave measured by a scalar pressure sensor (i.e., a microphone). Let u(p, t) denote the scalar value of the waveﬁeld at position p and time t. For a monochromatic source at frequency ω the waveﬁeld is

u(p, t) = α cos(ωt κ p + ϕ ) . (5.3) 0 − · 0

Suitable parametrization of α0 and ϕ0 makes the scalar wave model equivalent to 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 of Ns sensors on the surface of the earth positioned at locations pn n=1,...,Ns . We restrict our interest to small aperture arrays and work with{ a ﬂat} earth model, thus consider planar arrays. The signal at each sensor component is sampled at K instants t . In { k}k=1,...,K 101 general, each sensor measures a vector quantity. Let L be the total number of channels recorded by the array. In the case of scalar sensors, then simply L = Ns. (ℓ) Each measurement Yk is corrupted by additive white Gaussian noise and is modeled as

(ℓ) (ℓ) (ℓ) Yk = uk (θ) + Zk , (5.4)

(ℓ) 2 2 for each channel ℓ = 1,...,L where Zk (0, σℓ ). The noise variance , σℓ is, ∼ N (ℓ) in general, diﬀerent on each channel. The quantities uk (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 measurements is

L K (ℓ) (ℓ) 2 1 (yk uk (θ)) pY (y θ) = exp − , (5.5) 2 2σ2 | 2πσℓ − ℓ ! ℓY=1 kY=1 p (ℓ) where we grouped the measurements as Y = Yk ℓ=1,...L . { }k=1,...,K

Parameter Estimation Wavefield parameters can be found by using maximum likelihood (ML) estimation. ML estimation is a useful method for estimating the parameters of a statistical model. It is a widely-used estimation technique due to its broad applicability and to the optimal performances in many settings (Fisher, 1922). We refer the interested reader to Kay (1993) for additional details on ML estimation. An implementation this method for the estimation of wavefield parameters 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 measurements (5.5). We denote the LF by pY (y˜ θ), where y˜ are the observations and θ is the vector of the wavefield parameters| argument of the LF. We stress that the LF is a function of the model parameters θ while the measurements y˜ are fixed. An ML estimate θˆ of the true wavefield parameters θ˘ is found by maximizing the LF, i.e.,

θˆ = argmax pY (y˜ θ) . (5.6) θ | The LF can be thought of as a utility function which is legitimised by the statistical model of the observations. The point of maximum of the LF corresponds to the ML estimate of the parameters.

5.3 Sources of Error

In this section we describe in detail how the sensor positions aﬀect the performance of the ML parameter estimation. We make a distinction between two 102

10−3

10−4 ] 2 − 10−5 m

10−6 MSEE [

10−7 Threshold zone region

10−8 No information Asymptotic region

10−9 −20 −15 −10 −5 0 5 10 SNR [dB] Figure 5.2: An example of the MSEE of a ML estimator. The MSEE is depicted with a blue dashed line. In the no information region the MSEE is very large and constrained by the implementation of the algorithm. In the threshold region the occurrence of outliers keep the MSEE significantly larger than the CRB. At last, in the asymptotic region, the MSEE is well described by the CRB, which is shown with the black dashed line. types of errors: gross errors and fine errors (Athley, 2008). Array geometry affects 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 estimation of the wavenumber of an unknown wave with several different noise realization and for different SNRs. The SNR is defined as the ratio of signal power over 2 2 noise power, i.e., SNR = α0/2σ . Three operation regions of the estimator are recognized at different SNR ranges. The approximate extent of the regions is shown 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 are completely random and carry no information about the value of the parameter estimated. At larger SNR, there is a region called the threshold region. In this region, the estimated value may be often close to the true value, however the MSEE is still considerably large as gross estimation errors occur. Gross estimation errors are also known as global errors or outliers. Further increasing the SNR, we approach the asymptotic region. Fine estimation error occurs in this 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 also known as local errors. The abrupt increase in the MSEE below a certain SNR is known in literature as the threshold effect and is due to a transition from fine errors to gross errors (Van Trees, 2001).

Gross Errors Gross errors are due to the presence of local maxima (sidelobes) other than the true maximum in the LF (Athley, 2008). In this section, we establish the relationship between the LF and the array geometry. 103

Consider an array of Ns sensors at positions pn n=1,...,Ns . The spatial sampling pattern is given by a sum of Dirac delta located{ } at the sensor positions

Ns h(x, y) = δ(p p ) . (5.7) − n n=1 X Its Fourier transform is given by

−i(κx,κy )·p H(κx, κy) = h(x, y)e dp (5.8) R2 Z Ns = e−i(κx, κy )·pn , (5.9) n=1 X 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 gross errors and to establish a strategy to mitigate them. It is possible to verify that the function H(κ , κ ) always exhibits a global | x y | maximum at (κx, κy) = (0, 0). Other maxima also exist but, in general, are of 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 are given by

K g(t) = δ(t t ) (5.10) − k Xk=1 K G(ω) = g(t)e−iωtdt = e−iωtk , (5.11) R Z Xk=1 where tk k=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 expectation of 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 α˘0 and ϕ˘0 can be found explicitly as a function of the observations y˜ and of the wavevector κ. Therefore we write the LF of the observations as a function of κ as

pY (y˜ κ) = max pY (y˜ θ) , (5.12) | α0,ϕ0 | where the maximization is achieved by inserting into pY (y˜ θ) the ML estimates | αˆ0 and ϕˆ0. In Appendix 5.A, analytic expressions for αˆ0 and ϕˆ0 are given. Since we are interested only in the shape of the LF we compute its logarithm, and drop multiplicative and additive constants. We use the expectation operator E to obtain an indication about the average shape of the LF. After some manipulations,{·} explained in detail in Appendix 5.A, we obtain

2 E ln(pY (y κ)) G(ω ω˘)H(κ κ˘) , (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) is related to a translation of the Fourier transform of the sampling pattern. The quantity G( ) is of marginal importance concerning the occurrence of gross errors. Indeed· it is usually possible to sample the signal with sufficiently small sampling time and with enough samples. We observe that for ω exactly known G(ω ω˘) = G(0) is a real constant and thus does not change the shape of the LF. Hereafter− we will omit the factor G( ). The quantity H(κ) , or quantities closely related· to it, is known in literature with 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 (Gaffet et 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 around the local maxima of H(κ) . In order to reduce the occurrence of the gross errors 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 modeled as vector quantities. Therefore it is necessary to extend the findings concerning gross errors obtained for the scalar wave case to the vector wave case. When considering vector measurements, the PDF of the observations needs to be augmented with the contribution of all the sensor components. With the assumption of independent observations, this is achieved by increasing L (ℓ) in (5.5) and by choosing the appropriate wave model uk . Concerning gross errors, the shape of the LF of observations of Love and Rayleigh waves is influenced differently by the different components of each sensor but remains a function of the Fourier transform of the sampling pattern H(κ). The derivation of this relationships are detailed in Appendix 5.A and in this section we only present the final results.

Love Wave For Love waves as in (5.1), the LLF is related to H as

2 E ln(pY (y κ, ψ)) f (ψ, ψ˘) H(κ κ˘) , (5.14) { | } ∝ L | − | where f (ψ, ψ˘) = cos2(ψ ψ˘) . (5.15) L − From the previous expressions it is possible to understand that, similar to the scalar wave setting, the LLF for Love waves is directly related to the Fourier transform of the sampling pattern. The factor fL(ψ, ψ˘) inﬂuences the shape of the LLF as a function of azimuth.

Rayleigh Wave For Rayleigh waves as in (5.2), the LLF is related to H as 2 E ln(pY (y κ, ψ, ξ)) f (ψ, ψ,˘ ξ, ξ˘) H(κ κ˘) , (5.16) { | } ∝ R | − | 105 where 2 f (ψ, ψ˘, ξ, ξ˘) = sin ξ sin ξ˘cos(ψ ψ˘) + cos ξ cos ξ˘ . (5.17) R − We observe that for a ﬁxed ellipticity angle ξ the relationship is similar to the Love wave setting. For ﬁxed azimuth ψ and wavenumber κ, the occurrence of local maxima is described by trigonometric functions of ξ. We observe that the local maxima due to (5.17) are independent of the sensor positions and therefore are outside the scope of this work.

Fine Errors At high SNR, the performance of the ML estimator is well described by the CRB. 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 asymptotic region. To compute the CRB, we ﬁrst need to introduce the notion of Fisher information (FI). The FI conveys the amount of information about a statistical 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

2 ∂ ln pY (y θ) I(θ) = E | . (5.18) − ∂θ2 The matrix I is a square symmetric matrix with as many columns as the elements in the vector θ. The diagonal terms of the matrix correspond to the FI of each parameter in the parameter vector θ. These elements should be interpreted with care, as they disregard the uncertainty due to the other model parameters being unknown. The oﬀ-diagonal terms are sometimes referred to as cross-information terms. The information inequality (Cramér, 1946; Rao, 1945) states that the mean- squared estimation error (MSEE) of an unbiased estimator is lower bounded as

T E θˆ E θˆ θˆ E θˆ (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 θˆ and the right-hand side is the matrix inverse of the FIM. Following the information inequality, we are interested in the diagonal elements of I−1 as they provide a lower 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 wavenumber is lower bounded as

α2K Q2 (ψ) −1 E (ˆκ E κˆ )2 0 Q (ψ) ab . (5.20) − { } ≥ 2σ2 aa − Q (ψ) bb 106

The CRB is directly proportional to noise power σ2, inversely proportional to the amplitude of the wave α0 and to the number of samples K. The CRB depends on the sensor positions through Qaa, Qbb and Qab. We also observe that the wavenumber CRB is independent of the temporal frequency ω, thus we expect ﬁne errors to have comparable variance at any frequency. The quantities Qaa, Qbb, and Qab are called moment of inertia (MOI) of the array. These quantities are independent of array translations, but are in general dependent on the azimuth. We introduce the coordinate system (a, b), which is related to the coordinate system (x, y) as

a cos ψ sin ψ x = , (5.21) b sin ψ cos ψ y ! − ! ! where the angle of rotation is the azimuth ψ. Therefore a is the axis along the direction of propagation of the wave and b the axis perpendicular to it. The sensor positions in the rotated coordinate system are (an, bn) n=1,...,Ns . The MOI of the array in the coordinate system (a, b) are deﬁned{ as}

Ns Q (ψ) = (a a¯)2 (5.22) aa n − n=1 X Ns Q (ψ) = (b ¯b)2 (5.23) bb n − n=1 X Ns Q (ψ) = (a a¯)(b ¯b) , (5.24) ab n − n − n=1 X 1 Ns ¯ 1 Ns where a¯ = Ns n=1 an and b = Ns n=1 bn deﬁne the phase centre of the array. P P With reference to the CRB in (5.20), large Qaa and Qbb are desirable in order to reduce the CRB and thus the MSEE in the asymptotic region. Also a 2 small Qab 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 assumption which is of critical importance in practical applications. Moreover, it is possible to choose an array geometry such that Qab = 0 and thus eliminating the 2 term Qab/Qbb from (5.20). Some further remarks on the MOIs are given in Appendix− 5.B.

Love Wave and Rayleigh Wave In the vector case, the derivation of the CRB in Section 5.3 needs to be extended. However, for translational sensors the dependence of the CRB on array geometry remain similar to the one presented in (5.20) and therefore we do not review this aspect. A detailed analysis of the CRB of parameters of Love wave and Rayleigh wave measured with translational and/or rotational sensors is found in Maranò & Fäh (2013). 107

5.4 Problem Statement and Design Criterion

The aim of sensor placement is to improve the performance of parameter estimation, by an appropriate choice of the array geometry. In particular, we consider the setting where two design requirements are given: A budget of N sensors to be placed on a plane surface; • s An indication about the spatial bandwidth of the signals of interest. • In several applications, only a limited number of sensors is available. This is often the case for the analysis of seismic wavefields where each individual instrument can be quite expensive and/or difficult to install at the measurement site. Spatial bandwidth is defined by the spatial frequency content of the wavefield. Particularly important is the largest wavenumber present in the wavefield, denoted by κmax. Knowledge of spatial bandwidth is a reasonable assumption in many applications. In the seismic case the wavenumber is the goal of the estimation process and thus an exact knowledge of the signal bandwidth is not available. However, prior knowledge on the geology of the site or previous surveys 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 in order to reduce gross errors and fine errors, respectively. At low SNR, array design focuses on the design of the LF in order to reduce the height of local maxima and thus the occurrence of gross errors. We have shown as the Fourier transform of the spatial sampling pattern H(κ) plays a central role. At high SNR, array design targets the reduction of fine errors. This can be achieved by 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 errors is 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 it is necessary to reduce the amplitude of local maxima of the LF over a certain region of the κx κy-plane. We choose this region as an annulus defined by κmin and 2κmax, i.e., the region bounded by two concentric circles with radii κmin and 2κmax, respectively. We now explain the rationale behind this design choice. The circular symmetry of the considered region is motivated by the final application, the analysis of ambient vibration wavefields. In particular, by the fact 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 two resolvable signals is known in literature as the Rayleigh resolution limit (Van Trees, 2001). The exact Rayleigh resolution limit can be computed from a given sampling pattern and it is in general slightly different from κmin. In practice, the quantity κmin is also related to the smallest wavenumber that can be reliably estimated in field measurements. We recall that the LF is related to the Fourier transform of the sampling pattern H(κ) through a translation as in (5.13). We also observe that the ML 108 estimate of the wavevector is found as the largest value of the LF in a disk of radius κmax centred in the origin. A geometrical argument suggests that, in order to reduce gross errors, it is necessary to reduce the height of the local maxima of H(κ) in a disk of radius 2κmax. We illustrate| this| argument with an example. In Fig. 5.3, we show graphically how the LF is related to a translation of the Fourier transform of the sampling pattern. Fig. 5.3(a) shows the magnitude of the Fourier transform of an optimised sampling pattern. The two magenta circles have radius κmin and 2κmax, respectively. In this example κmin = 1/4 and κmax = 1. We observe the presence of a global maximum at (0, 0) inside the inner circle, that the local maxima in the annulus region are small in amplitude, and that outside the outer 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 of the expected LLF is located at (0.5, 0). Observe that the shaded region of Fig. 5.3(b) is exactly the Fourier transform of the sampling pattern shown in Fig. 5.3(a). Fig. 5.3(c) depicts the expected LLF of a wave with wavenumber κ˘ = (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 parameterk k 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 design since κ˘ > κmax. The global maximum of the expected LLF corresponds to a localk maximak of H(κ) outside the minimization region. As results, the ML estimate of the wavevector| | is not corresponding to the true wavevector. We formulate the following minimization program, aimed at minimizing the largest value of H(κ) over the annulus defined by κ κ 2κ as | | min ≤ k k2 ≤ max

min max H(κ, p1, p2,..., pNs ) (5.25a) p1,p2,...,pNs κ s. t. κ κ 2κ , (5.25b) min ≤ k k2 ≤ max where we emphasized the dependence of H on sensor positions. In this optimization problem, the minimization variables are the sensor positions. The target of the minimization, maxκ H(κ, p1, p2,..., pNs ) , is a function of the sensor positions p . To the best of our knowledge, the use of such { n}n=1,...,Ns cost function is new to the seismological community. This minimization problem becomes harder as the extent of the annulus is increased. 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 pn n=1,...,Ns and on the azimuth ψ. Indeed, in general, diﬀerent MSEE values{ are} expected for diﬀerent DOAs of the incoming wave. We choose to maximize the inverse of the CRB for the DOA leading to the 109

2 2

1 1 [1/m] [1/m] y y κ κ 0 0

−1 −1 Wavenumber Wavenumber

−2 −2

−2 −1 0 1 2 −1 0 1 2 Wavenumber κx [1/m] Wavenumber κx [1/m] (a) Fourier transform of an optimised sam- (b) Expected LLF, κ˘ = (0.5, 0). pling pattern.

2 2

1 1 [1/m] [1/m] y y κ κ 0 0

−1 −1 Wavenumber Wavenumber

−2 −2

−1 0 1 2 3 0 1 2 3 Wavenumber κx [1/m] Wavenumber κx [1/m] (c) Expected LLF, κ˘ = (1, 0). (d) Expected LLF, κ˘ = (1.5, 0).

Figure 5.3: Illustration of the relationship between the magnitude of the Fourier transform of the sampling pattern and expected LLFs of diﬀerent waves. worst performance

2 Qab(ψ, p1,...) max min Qaa(ψ, p1,..., pNs ) . (5.26) p1,p2,...,pNs ψ − Q, (ψ p ,...) bb 1 Such maximization problem is unbounded. Indeed, it is possible to verify that a uniform circular array (UCA) with suﬃciently large radius can make the objective function arbitrarily large. Moreover, an array with a very large aperture could void the assumption of planar wavefronts.

Discussion Elaborating on the two distinct sources of errors discussed in Section 5.3, we showed how two different sensor placement criteria exist. Depending on the application, 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 order to account for both types of errors. For ambient vibration wavefields, typically the waves have very small amplitudes compared to the level of the background noise. Therefore we choose to primarily focus on improving performance at low SNR and thus on the reduction of gross errors. Although we choose not to address fine errors explicitly, we consider arrays with Qab = 0. This additional constraint is beneficial for reducing the CRB of (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 complexity that can always ﬁnd the global optimal solution to the program in (5.25). The objective function (5.25a) has many local minima and any solution found by, for example, a gradient descent method will strongly depend on the initial starting point. In this section, we propose an algorithm using a mixed integer program (MIP), which is shown empirically to be eﬀective. We also compare it with two other possible approaches addressing the same optimization problem.

Mixed Integer Program The program in (5.25) is diﬃcult to solve because the minimization variables

pn n=1,...,Ns are arguments of the complex exponentials. To overcome this limitation,{ } instead of representing the sensor positions with continuous variables we consider a ﬁnite number of possible sensor positions. We represent the discrete sensor positions with the binary vector x 0, 1 N . An analogous minimization problem in this new variable is ∈ { }

min Fx (5.27a) x k k∞ s. t. Asx = bs (5.27b) x 0, 1 N , (5.27c) ∈ { } where v ∞ = max( v1 , v2 ,..., vN ). Thek vectork 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 possible choice of N feasible spatial locations for positioning the Ns available sensors. The linear operator F : RN CM computes the two dimensional Fourier transform of the array positions,→ restricted to M discrete spatial frequencies of interest. The operator F can be thought as a discretized version of H(κ). The

M frequencies lie in the annulus deﬁned by κmin κ 2 2κmax . Fig. 5.4(b) shows a possible choice of the M frequencies computed≤ k k ≤ by F. Observe that because of the symmetry H(κ) = H( κ) the choice of the M frequencies | | | − | can be limited only to half of the κx κy-plane. We choose to arrange both the N possible sensor positions and and the M spatial frequencies on circles around the origin, as in Fig. 5.4. This arbitrary choice is supported by the radial symmetry of the problem as expressed 111

0.5 1 [1/m] y κ

0 0 y [m]

−1 Wavenumber −0.5

−2 −0.5 0 0.5 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] (a) Possible N locations available in the (b) Position of the M spatial frequencies x y-plane for the placement of Ns sensors. in the κx κy-plane. The magenta circles depict κmin and 2κmax.

Figure 5.4: A possible choice of the N spatial locations and of the M spatial frequencies used in the construction of the operator F. in (5.25). Concerning the sensor positions, this choice also makes it easier to enforce numerically any constraint on the MOIs. The equalities in (5.27b) enforce a number of linear constraints. The number of sensors is enforced to be equal to Ns. The vector x has exactly Ns elements equal to one, corresponding to the sensor positions. The remaining elements are zero. As an extension, using linear equalities is also possible to enforce the presence or the absence of a sensor in a speciﬁc position. In (5.27b) we also enforce the linear constraint Qab = 0. This constraint al- 2 lows to eliminate the term Qab/Qbb from the expression of the CRB in (5.20), thus lowering the CRB. It− also ensures that the array performance, in terms of ﬁne errors, would have no preferential direction, i.e., Qaa and Qbb are constant for every azimuth. Comparing the original problem in (5.25) and the discretized formulation in (5.27), we observe that the discretization of the sensor positions causes the optimal value of (5.25) to be smaller than or equal to the optimal value of (5.27). This is especially important when the vector x has small dimension N and the possible sensor positions are coarse. Observe that the objective function (5.27a) is a convex function of the minimization variable x. However, due to the binary constraint (5.27c) the minimization is an integer programming problem. Integer programs are generally considered to be NP-hard, i.e. there is no polynomial time algorithm to solve them. We relax the optimization problem in (5.27) to make it more tractable for implementation. We replace the convex objective function with a linear objective function as

Re Fx Fx | | . ∞ F k k → Im x ! | | ∞

112

This modiﬁcation enables us to formulate the problem as a MIP with linear objective function and linear constraints

min y (5.28a) y

s. t. Asx = bs (5.28b) Re F Im F   x 1y (5.28c) Re F  −   Im F   −  y R  (5.28d) ∈ x 0, 1 N , (5.28e) ∈ { } where 1 is a vector of ones of size 4M 1 and denotes element-wise . In this program there are N binary variables× and one continuous real variable.≤ This can be addressed using general purpose MIP algorithms (Gurobi Optimization, Inc., 2013). It is in theory possible to ﬁnd the optimal solution to (5.28) using the branch and bound algorithm (Land & Doig, 1960). However, for a large number of possible sensor positions N, ﬁnding the optimal solution becomes not 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 solutions using some random search pattern and there is no guarantee to ﬁnd the optimal solution. In our implementation we do not enforce any constraint on the MOIs.

The constraint on the Qab = 0 is nonlinear in the variables pn n=1,...,Ns . We do not enforce such constraint in the considered GA technique{ } since we observed that it makes the minimization considerably harder.

Uniform Circular Array In addition, we compare with the best uniform circular array (UCA). In an UCA sensors are uniformly spaced on a ring of radius r. A line search over the possible r allows us to obtain the radius of the best UCA for given κmin, κmax, and Ns

min max H(κ, p , p ,..., p ) (5.29a) r κ 1 2 Ns s. t. p = r(cos(2πn/N ), sin(2πn/N )) (5.29b) n s s κ κ 2κ . (5.29c) min ≤ k k2 ≤ max The objective function is the same of the original problem (5.25). Sensors are restricted to be uniformly spaced on a circle of radius r. We state again that all UCA with N 3 have Q = 0. Moreover Q and s ≥ ab aa Qbb 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 techniques and we quantify the impact different geometries have on the estimation problem. In Section 5.6, the outcomes of the different array design methods are 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 is analyzed 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; and iii) 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), the goodness of a solution is quantiﬁed with the amplitude of the largest local maxima of H(κ) compared to the central maximum H(0, 0) . Therefore, we consider the| quantity| | |

H(κ) 2 Hmax = max | | (5.30) κ 2 Ns ! for κmin κ 2 2κmax . Observe that the quantity Hmax is smaller than 1, since the≤ largest k k value≤ of H(κ) is N . | | s Fig. 5.5 shows the value of Hmax achieved with different design techniques for different number of sensors Ns and κmax/κmin . In general, the value of Hmax decreases for increasing number of sensors Ns. In particular, such decrease is steeper for few sensors. This indicates that the marginal benefit of an additional sensor 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, and 1/6, respectively. A large value of κmax/κmin corresponds to a larger extension of the annulus involved in the minimization. For larger values of κmax/κmin the minimization problem is harder and the Hmax values found are indeed larger. In Fig. 5.5, we observe that the MIP technique typically achieves values of Hmax smaller than the other design techniques, for all the considered κmax/κmin. We observed that the quality of the MIP solutions and the time necessary to 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 sensor locations on each concentric circle has to be related to Ns. One motivation is that due to the Qab = 0 constraint and the arrangement of the possible sensor locations, the MIP solutions exhibit sensors placed on concentric circles. Therefore certain choices of the number of possible locations on each circle may turn the MIP problem infeasible, and other choices may be more convenient. 114

For certain choices of Ns and arrangement of the possible spatial locations the globally optimal solution was found. For larger Ns and N the algorithm was terminated after a given time limit. The GA technique is able to consistently decrease Hmax for increasing number of sensors. In contrast to the MIP technique, there is no unfavorable sensor number when the GA technique is used. The GA solutions are in general worse than the MIP solutions in terms of Hmax. In terms of minimization, an intrinsic advantage of the GA technique is that the minimization variables pn n=1,...,Ns are continuous, this may help the algorithm to optimize the solution,{ } at least locally. The GA solution presented is the best out of hundreds of runs performed with diﬀerent initialization of the GA algorithm. The UCAs technique performs similarly or worse than the other techniques depending 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 of sensors above 10. For κmax/κmin equals to 4 and 6, the UCA is often largely outperformed by the other two design techniques.

Fine Errors At high SNR, the performance of the ML estimator is well characterized by the CRB. From (5.20) it is clear how the CRB performance depends on the azimuth. We deﬁne the quantity Qmin, related to the CRB at the azimuth ψ exhibiting the worst performance as

2 Qab(ψ) Qmin = min Qaa(ψ) . (5.31) ψ − Q (ψ) bb

Note that Qmin is not explicitly taken into account in any of the strategies considered for array design. However, the MIP and the UCA geometries are guaranteed to have Qab = 0 for any azimuth. Fig. 5.6 depicts the value of Qmin for the same arrays considered in Fig. 5.5(b). As expected, the value of Qmin is generally an increasing function of the number of sensors.

Array Geometry Example array geometries are depicted in Fig. 5.7 and Fig. 5.8. The array geometries 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 of Fig. 5.7(b) exhibit a completely irregular pattern since the sensor positions are unconstrained. The UCA array is shown in Fig. 5.7(c). A spiral shaped array is shown in Fig. 5.8(a). From the large width of the central lobe, a large MSEE at high SNR is to be expected for the spiral array. A cross shaped array is shown in Fig. 5.8(b). Due to the regular geometry, the cross array exhibits many maxima having the same magnitude as the central lobe. In these two latter arrays, sensor spacings are chosen arbitrarily and the arrays are not designed to follow the proposed criteria. 115

0.8 MIP 0.7 GA 0.6 UCA

0.5 max 0.4 H

0.3

0.2

0.1

0 4 6 8 10 12 14 16 18 20 Number of sensors Ns

(a) κmin = 1/2 and κmax = 1.

0.8 MIP 0.7 GA 0.6 UCA

0.5 max 0.4 H

0.3

0.2

0.1

0 4 6 8 10 12 14 16 18 20 Number of sensors Ns

(b) κmin = 1/4 and κmax = 1.

0.8 MIP 0.7 GA 0.6 UCA

0.5 max 0.4 H

0.3

0.2

0.1

0 4 6 8 10 12 14 16 18 20 Number of sensors Ns

Figure 5.5: Values of Hmax achieved with different array design strategies for different number of sensors. Different κmax/κmin are considered. A small Hmax is desirable in order to reduce the occurrence of gross errors. 116

30 MIP GA 25 UCA

min 15 Q

0 4 6 8 10 12 14 16 18 20 Number of sensors Ns

Figure 5.6: Values of Qmin obtained from diﬀerent array design strategies for diﬀerent number of sensors and κmax/κmin = 4. A large Qmin is desirable in order 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 be found in Fig. 5.5(b) and Fig. 5.6, respectively. The symmetry of the MIP array and the irregular pattern of the GA array are also observed for other sensor numbers and values of κmax/κmin. This fact suggests that the actual deployment in the ﬁeld of a MIP array may be signiﬁcantly easier that the deployment of a GA array. Additional array layouts obtained 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 actual estimation problem we resort to the analysis of the MSEE using Monte Carlo simulations. Arrays obtained with the sensor placement techniques considered in the previous section are compared. Moreover, we compare with an array having sensors arranged on a spiral and with another array with sensors arranged on a cross. These two latter arrays are designed without following the design criterion proposed in this work. The MSEE is computed as follows. A wavevector is drawn randomly from the uniform distribution having as support a disk with radius κmax. For each considered array geometry and SNR, random noise is added to the wavefield as in (5.5). The ML estimation method of Maranò et al. (2012) is used to estimate the parameters of the wave and the MSEE is obtained by repeating the 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 wavenumber for different array geometries. The MIP, GA, and UCA arrays are designed 117

1.4 2

0.7 [1/m] 1 y κ

0 0 y [m]

−0.7 −1 Wavenumber −2 −1.4

−1.4 −0.7 0 0.7 1.4 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] (a) MIP array.

3.2 2

2.4

[1/m] 1 y 1.6 κ 0 y [m] 0.8 −1 0 Wavenumber −2 −0.8

−2.4 −1.6 −0.8 0 0.8 1.6 2.4 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] (b) GA array.

1 2

0.5 [1/m] 1 y κ

0 0 y [m]

−1 −0.5 Wavenumber −2 −1

−1 −0.5 0 0.5 1 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] (c) UCA array.

Figure 5.7: Array geometries h(x, y) and the corresponding H(κ , κ ) 2. The | x y | arrays are obtained by choosing Ns = 14, κmin = 0.25, and κmax = 1. The magenta circles delimit the annulus κ < κ < 2κ . min k k2 max 118

0.9

2 0.6

[1/m] 1

0.3 y κ 0 0 y [m]

−0.3 −1 Wavenumber −0.6 −2

−0.9 −0.9 −0.6 −0.3 0 0.3 0.6 0.9 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] (a) Spiral array.

1.8 2

0.9 [1/m] 1 y κ

0 0 y [m]

−1 −0.9 Wavenumber −2 −1.8

−1.8 −0.9 0 0.9 1.8 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] (b) Cross array.

2 Figure 5.8: Array geometries h(x, y) and the corresponding H(κx, κy) . The magenta circles delimit the annulus κ < κ < 2κ . | | min k k2 max

with κmax/κmin = 4. In general, the MSEE decreases as the SNR increases and it achieves the CRB for suﬃciently large SNR. Fig. 5.9(a) shows the MSEE for an array of Ns = 7 sensors. The MIP, the GA, and the UCA exhibit similar performances in the threshold region. At high SNR, in the asymptotic region, certain arrays perform better than others. We observe 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. The cross array does not achieve its CRB because the local maxima of the LLF are indistinguishable from the right maximum. This ambiguity is due to the regular array geometry, cf. Fig. 5.8(b). Fig. 5.9(b) shows the MSEE for an array of Ns = 14 sensors. The geometry of these arrays is shown in Fig. 5.7. The MIP, the GA, and the UCA arrays exhibit similar performance both at low and high SNR. The spiral array and the cross array again exhibit the larger MSEE. In Fig. 5.10 the MSEE of the ML estimate of the wavenumber for diﬀerent 119

10−1 ] 2 −2

− 10

10−3 MIP MSEE [m GA UCA 10−4 Spiral Array Cross Array

−35 −30 −25 −20 −15 −10 −5 SNR [dB]

(a) Ns = 7.

10−1

−2

] 10 2 −

10−3 MIP MSEE [m GA 10−4 UCA Spiral Array Cross Array

−35 −30 −25 −20 −15 −10 −5 SNR [dB]

(b) Ns = 14.

Figure 5.9: Comparison of the MSEE of the wavenumber obtained with different array geometries. The CRBs are also depicted with black thin dotted lines. array geometries and number of sensors is depicted. Only the MIP geometries, with κmax/κmin = 4, are considered for different values of the SNR. As expected, the MSEE decreases when the number of sensors is increased. For SNR = 25 dB, the MSEE is constantly well above the CRB because the SNR is low and− the estimator operates in the threshold region, thus gross errors occur. For SNR = 20 dB, the MSEE achieves the CRB for a sufficiently large number of sensors.− At last, a SNR of 15 dB is sufficient for the estimator to operate in the asymptotic region for any− number of sensors. In Fig. 5.10, it is also observed that the marginal benefit of an additional sensor also depends on the SNR. When the estimator operates in the asymptotic region (e.g. SNR = 15 dB), most of the MSEE reduction is achieved with the first 8 sensors. However− for SNR = 20 dB, the MSEE reduction − is still significant until Ns = 11 because the threshold effect of the estimator. 120

10−1 SNR -25 [dB] SNR -20 [dB] SNR -15 [dB] ] −2 2 10 −

10−3 MSEE [m

10−4

4 6 8 10 12 14 16 18 20 Number of sensors Ns Figure 5.10: Comparison of the MSEE of the wavenumber obtained with different array geometries having a diﬀerent number of sensors. The CRBs are also depicted with black thin dotted lines.

These considerations may also vary for diﬀerent κmax/κmin as suggested by Fig. 5.5.

5.7 Conclusions

In this paper, we described in detail the occurrence of gross errors and fine errors in the estimation of parameters of Love waves and Rayleigh waves. We derived a relationship between the Fourier transform of the sampling pattern and the average likelihood function of the observations. Sensor placement criteria for the reduction of gross errors and fine errors are then proposed. An array design algorithm employing the proposed criterion with linear objective function and linear constraints is formulated as a mixed integer program. The proposed algorithm is compared with the genetic algorithm technique and with the uniform circular array technique using the same design criterion. The proposed algorithm achieves superior design for most choices of the sensor number and spatial frequency requirements. In addition, the proposed sensor placement algorithm generates arrays composed of simple regular geometries and are thus suitable for field deployment. The mean-squared estimation errors of the maximum likelihood estimator using different array geometries are compared by means of Monte Carlo simulations. We find that the proposed design criterion is suitable for the optimised placement of seismic sensors for the analysis of Love waves and Rayleigh waves. We show that the sensor placement techniques considered can greatly reduce the mean-squared estimation error when compared with non optimised arrays. We emphasize that prior knowledge of the spatial frequency support of the 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 is desirable. Future extensions of this work will address the presence of physical obstructions limiting the available space to deploy sensors. 121

5.8 Acknowledgments

This work is supported in part by the Swiss Commission for Technology and Innovation 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 and Dr. 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 the | Fourier transform of the sampling pattern H(κx, κy).

Scalar case We consider the scalar wave equation of (5.3) and the measurement model of (5.4). We parametrize the wave equation as a function of α1, α2 instead of α , ϕ using { } { 0 0} α1 = α0 cos ϕ0 α = α sin ϕ . 2 − 0 0 This parametrization is equivalent because the transformation between the two parameters pairs is bijective. (n) The measurement Yk , at the time instant tk and location pn is therefore

(n) (n) Yk = u(pn, tk) + Zk = α cos(ωt κ p + ϕ ) + Z(n) 0 − · 0 k = α cos(ωt κ p) + α sin(ωt κ p) + Z(n) . 1 − · 2 − · k For a given κ, the ML estimates αˆ1 and αˆ2 can be found analytically and are

Ns K 2 αˆ = Y (n) cos(ωt κ p ) 1 N K k k − · n s n=1 X Xk=1 Ns K 2 αˆ = Y (n) sin(ωt κ p ) , 2 −N K k k − · n s n=1 X Xk=1 and their expected values are

Ns 2 K E αˆ (˘α cos(˘ωt κ˘ p ) cos(ωt κ p ) { 1} ≈ N K 1 k − · n k − · n s n=1 X Xk=1 +˘α sin(˘ωt κ˘ p ) cos(ωt κ p )) 2 k − · n k − · n Ns 2 K E αˆ (˘α cos(˘ωt κ˘ p ) sin(ωt κ p ) { 2} ≈ −N K 1 k − · n k − · n s n=1 X kX=1 +˘α sin(˘ωt κ˘ p ) sin(ωt κ p )) , 2 k − · n k − · n 122 where the superscript˘denotes the true value of the parameter and the lack of superscript denotes search parameters, e.g., the argument of the LF. When κ = κ˘, then E αˆ1 α˘1 and E αˆ2 α˘2. The symbol denotes an approximation. We used{ the} following ≈ trigonometric{ } ≈ approximations≈

K K cos2(ωk + γ) ≈ 2 kX=1 K K sin2(ωk + γ) ≈ 2 Xk=1 K sin(ωk + γ) cos(ωk + γ) 0 , ≈ Xk=1 2πm Z which are valid for ω being not near 0 or 1/2 and are exact when ω = K m (see example 3.14 in Kay (1993) or Stoica et al. (1989)). ∈ The second moments of αˆ1 and αˆ2 are

E 2 E 2 2 2 αˆ1 αˆ1 + σ { } ≈ { } NsK and

E 2 E 2 2 2 αˆ2 αˆ2 + σ , { } ≈ { } NsK respectively. From the previous derivations, it is clear to see the dependence on κ of αˆ1(κ) and αˆ2(κ). Therefore, given observations y˜ and plugging αˆ1 and αˆ2 into the PDF (5.5), it is possible to rewrite the LF of the observation as a sole function of the κ

pY (y˜ κ) = max pY (y˜ θ) | α0,ϕ0 |

= pY (y˜ (ˆα (κ), αˆ (κ), κ)) . | 1 2 In order to investigate the shape of the LF, we take the natural logarithm of the LF and drop all the additive and multiplicative constants not depending on κ

Ns K (n) (n) 2 ln(pY (y κ)) (Y u (θ)) | ∝ − k − k n=1 X Xk=1 Ns K 2 = u(n)(θ˘) + Z(n) u(n)(θ) , − k k − k n=1 X Xk=1 (n) where again we distinguish between the true and unknown wavefield u˘k = (n) ˘ (n) (n) uk (θ) and the wavefield as function of the search parameters uk = uk (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

Ns K 2 (n) (n) (n) E ln(pY (y κ)) E u˘ + Z u { | } ∝ − k k − k ( n=1 ) X kX=1 E αˆ 2 + E αˆ 2 ∝ { 1} { 2} E αˆ + i E αˆ 2 ≈ | { 1} { 2}| 2 Ns K (n) −i(ωtk −κ·pn) = u˘k e n=1 X kX=1 2 = α˘ e−iϕ˘0 G(ω ω˘)H(κ κ˘) . 0 − − E Ns K (n) 2 E Ns K (n) 2 Where the quantities n=1 k=1(˘uk ) and n=1 k=1(Zk ) were dropped since they are constants with respect to κ. Moreover we used nP P o nP P o

Ns K Ns K u(n)u˘(n) = (ˆα cos(ωt κ p ) +α ˆ sin(ωt κ p )) k k 1 k − · n 2 k − · n n=1 n=1 X Xk=1 X kX=1 (˘α cos(˘ωt κ˘ p ) +α ˘ sin(˘ωt κ˘ p )) · 1 k − · n 2 k − · n N L s (ˆα E αˆ + αˆ E αˆ ) ≈ 2 1 { 1} 2 { 2} and

Ns K Ns K (u(n))2 = (ˆα cos(ωt κ p ) +α ˆ sin(ωt κ p ))2 k 1 k − · n 2 k − · n n=1 n=1 X Xk=1 X kX=1 N L s αˆ2 +α ˆ2 . ≈ 2 1 2 Vector case In the vector case, each sensor component may experience a diﬀerent amplitude scalings or phase delay on each component, as shown in (5.1) and (5.2). (ℓ) (ℓ) The amplitude scalings β ℓ=1,...,L and the phase delays γ ℓ=1,...,L are, in general, functions of{ the waveﬁeld} parameters θ except for{ α}and ϕ. The (ℓ) measurement Yk , from the ℓ-th channel, at the time instant tk and at position pℓ is

(ℓ) (ℓ) Yk = u(pℓ, tk) + Zk = α β(ℓ) cos(ωt κ p + ϕ + γ(ℓ)) + Z(ℓ) 0 k − · ℓ 0 k = α β(ℓ) cos(ωt κ p + γ(ℓ)) + α β(ℓ) sin(ωt κ p + γ(ℓ)) + Z(ℓ) . 1 k − · ℓ 2 k − · ℓ k (ℓ) (ℓ) With a suitable parametrization of β , γ ℓ=1,...,L, this model is able to cap- ture all the amplitude scaling and phase{ delays} as in the seismic wave equations of (5.1) and (5.2). Similar to the scalar case, it is possible to obtain analytic expressions of both ML estimates αˆ1 and αˆ2 as 124

L K (ℓ) (ℓ) (ℓ) 2 ℓ=1 k=1 Yk β cos(ωtk κ pℓ + γ ) αˆ1 = − · . K L (ℓ) 2 P P ℓ=1(β )

Here and in what follows, we omit theP formulas concerning αˆ2 because of their similarity with the derivations of αˆ1. Details on the estimation of αˆ1 and αˆ2 in this setting can be found in Reller et al. (2011). The ﬁrst and the second moments of αˆ1 are

L K 2 1 (ℓ) (ℓ) (ℓ) E αˆ1 β α˘1β˘ cos(˘ωtk κ˘ p +γ ˘ ) { } ≈ K L (ℓ) 2 − · ℓ ℓ=1(β ) Xℓ=1 Xk=1 +˘αPβ˘(ℓ) sin(˘ωt κ˘ p +γ ˘(ℓ)) cos(ωt κ p + γ(ℓ)) 2 k − · ℓ k − · ℓ E 2 E 2 2 2 αˆ1 αˆ1 + σ . { } ≈ { } NsK As for the scalar case, we are interested in the average shape. Therefore we take the logarithm of the LF, take the expectation, and drop multiplicative and additive constants to get

2 2 E ln(pY (y θ)) E αˆ + E αˆ { | } ∝ { 1} { 2} L K (n) (ℓ) 2 (ℓ) −i(ωtk −κ·pℓ+γ ) ℓ=1 k=1 β u˘k e L . ≈ (β(ℓ))2 P P ℓ=1

P We observe that, in the wave models of (5.1) and (5.2) the amplitude scalings and the phase delays are identical on the same component of diﬀerent sensors. We denote with βx, γx, βy, γy, and βz, γz the scalings and delays on L (ℓ) 2 the x,y, and z components, respectively. Moreover the quantity ℓ=1(β ) is constant with respect to κ. The expression can be further simpliﬁed by grouping identicalP scalings

i(˘γx−γx) i(˘γy −γy ) i(˘γz −γz ) E ln(pY (y θ)) (β β˘ e + β β˘ e + β β˘ e ) { | } ∝ x x y y z y 2 αe˘ −iϕ˘G(ω ω˘)H(κ κ˘) . · − −

5.B Some Remarks on the MOIs of an Array

In this section, we show that any array with Qab = 0 has uniform properties, in term of ﬁne errors, at any azimuth. In addition, we also show that all uniform circular 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). P For any array, if Qab = 0 then Qaa and Qbb are constant. Consider the array with sensor positions

pn = (xn, yn) = rn (cos ψn, sin ψn) . 125

Using (5.22), (5.23), (5.24), and (5.21) the MOIs are found to be

Ns Ns Q (ψ) = a2 = r2 cos2(ψ ψ) aa n n n − n=1 n=1 X X Ns Ns Q (ψ) = b2 = r2 sin2(ψ ψ) bb n n n − n=1 n=1 X X Ns Ns Q (ψ) = a b = r2 cos(ψ ψ) sin(ψ ψ) . ab n n n n − n − n=1 n=1 X X The derivatives of Qaa and Qbb with respect to ψ are

Ns ∂Q (ψ) aa = 2r2 cos(ψ ψ) sin(ψ ψ) ∂ψ n n − n − n=1 X Ns ∂Q (ψ) bb = 2r2 sin(ψ ψ) cos(ψ ψ) . ∂ψ − n n − n − n=1 X From the previous expressions we conclude that if Qab(ψ) = 0 for every ψ, then Qaa and Qbb are constant for every ψ.

All UCAs have Qab = 0. A UCA is composed of Ns sensors equally spaced on a circle of radius r. Sensor positions are given by

2πn 2πn p = (x , y ) = r cos , sin . n n n N N s s Using (5.24) and (5.21), Qab for an UCA can be computed as

Qab(ψ) = anbn n=1 X Ns 2πn 2πn = r2 cos ψ sin ψ N − N − n=1 s s X Ns r2 2πn = sin 2 ψ 2 N − n=1 s X = 0 , for any ψ and Ns 3. The last equality can be proved using de Moivre’s formula (Abramowitz≥ & Stegun, 1964).

All UCAs have Qaa and Qbb constants at every azimuth. This fact follows directly from the two previous considerations.

5.C MIP Array Layouts

In this section, we provide the layouts found using the MIP method for different number of sensors. We explain how it is possible to use the presented 126 geometries by considering the largest wavenumber in the wavefield in practical applications. Using the scaling property of the Fourier transform it is possible to stretch or compress the array layout according to the largest wavenumber in the wavefield and therefore adapt the Fourier transform of the sampling pattern to the actual frequency content of the wavefield. The array layouts presented in this section ⋆ are obtained using κmax = 1 and different values of κmin. Let κmax be the ⋆ ⋆ largest wavenumber in the wavefield. Let h (x, y) and H (κx, κy) denote the desired sampling pattern and its Fourier Transform, respectively. They are related to the h(x, y) and H(κx, κy) provided in Fig. 5.11 and Fig. 5.12 as

⋆ ⋆ ⋆ h (x, y) = h (xκmax, yκmax) 1 κ κ H⋆(κ , κ ) = H⋆ x , y . x y κ⋆ κ⋆ κ⋆ max max max ⋆ ⋆ ⋆ The eﬀective κmin is also changed as κmin = κminκmax. Choice of which κmin to choose should be related in particular to the smallest wavenumber of interest in the analysis. In fact, we observe how a small κmin leads to arrays with larger aperture. ⋆ In practice, stretching the spatial sampling pattern by 1/κmax allows us to ⋆ ⋆ obtain a H (κx, κy) with local maxima minimized in the annulus κminκmax ⋆ ≤ κ 2 2κmax. k kThe≤ electronic supplement to this article provides tables with coordinates of each array. The supplement is found in Appendix D. 127

0.4 2 2 2 1.2 1.2

0 0 0 0

0 0

−2 −2 −2 −0.4

−2 0 2 −2 0 2 −2 0 2 −0.4 0 0.4 −1.2 0 1.2 −1.2 0 1.2

(a) Ns = 7, κmax/κmin = 2. (b) Ns = 7, κmax/κmin = 4. (c) Ns = 7, κmax/κmin = 6.

0.5 1.1 1.4 2 2 2

0 0 0 0 0 0

−2 −2 −2

−0.5 −1.1 −1.4 −2 0 2 −2 0 2 −2 0 2 −0.5 0 0.5 −1.1 0 1.1 −1.4 0 1.4

(d) Ns = 8, κmax/κmin = 2. (e) Ns = 8, κmax/κmin = 4. (f) Ns = 8, κmax/κmin = 6.

1.1 2 2 2 0.6

0 0 0 0 0 0

−2 −2 −2 −1.5 −1.1 −0.6 −2 0 2 −2 0 2 −2 0 2 −0.6 0 0.6 −1.1 0 1.1 −1.5 0 1.5

(g) Ns = 9, κmax/κmin = 2. (h) Ns = 9, κmax/κmin = 4. (i) Ns = 9, κmax/κmin = 6.

1.3 0.6 2 2 2 1.9

0 0 0 0 0 0

−2 −2 −2 −1.3 −0.6 −2 0 2 −2 0 2 −2 0 2 −0.6 0 0.6 0 1.3 −1.9 0 1.9

(j) Ns = 10, κmax/κmin = 2. (k) Ns = 10, κmax/κmin = 4. (l) Ns = 10, κmax/κmin = 6.

1.1 2 2 1.8 2 0.6

0 0 0 0 0 0

−2 −2 −2

−1.1 −0.6 −2 0 2 −2 0 2 −1.8 −2 0 2 −0.6 0 0.6 −1.1 0 1.1 −1.8 0

(m) Ns = 11, κmax/κmin = 2. (n) Ns = 11, κmax/κmin = 4. (o) Ns = 11, κmax/κmin = 6.

1.3 2.2 2 2 2

0 0 0 0 0 0

−2 −2 −2 −0.8 −1.3 −2.2 −2 0 2 −2 0 2 −2 0 2 −0.8 0 0.8 −1.3 0 −2.2 0

(p) Ns = 12, κmax/κmin = 2. (q) Ns = 12, κmax/κmin = 4. (r) Ns = 12, κmax/κmin = 6.

Figure 5.11: Array layouts found with the MIP method for Ns = 7,..., 12. 128

1.3 0.7 2 2 2

0 0 0 0 0 0

−2 −2 −1.9 −2 −1.3 −0.7 −2 0 2 −2 0 2 −2 0 2 0 0.7 −1.3 0 1.3 −1.9 0 1.9

(a) Ns = 13, κmax/κmin = 2. (b) Ns = 13, κmax/κmin = 4. (c) Ns = 13, κmax/κmin = 6.

0.8 1.5 1.9 2 2 2

0 0 0 0 0 0

−2 −2 −2

−0.8 −1.5 −1.9 −2 0 2 −2 0 2 −2 0 2 −0.8 0 0.8 −1.5 0 1.5 −1.9 0 1.9

(d) Ns = 14, κmax/κmin = 2. (e) Ns = 14, κmax/κmin = 4. (f) Ns = 14, κmax/κmin = 6.

2.1 0.7 2 2 2

0 0 0 0 0 0

−2 −2 −2 −1.5 −0.7 −2.1 −2 0 2 −2 0 2 −2 0 2 −0.7 0 0.7 −1.5 0 1.5 −2.1 0 2.1

(g) Ns = 15, κmax/κmin = 2. (h) Ns = 15, κmax/κmin = 4. (i) Ns = 15, κmax/κmin = 6.

1.6 0.8 1.8 2 2 2

0 0 0 0 0 0

−2 −2 −2 −1.6 −0.8 −1.8 −2 0 2 −2 0 2 −2 0 2 −0.8 0 0.8 −1.6 0 1.6 −1.8 0 1.8

(j) Ns = 16, κmax/κmin = 2. (k) Ns = 16, κmax/κmin = 4. (l) Ns = 16, κmax/κmin = 6.

0.8 1.4 2.4 2 2 2

0 0 0 0 0 0

−2 −2 −2

−0.8 −1.4 −2.4 −2 0 2 −2 0 2 −2 0 2 −0.8 0 0.8 −1.4 0 1.4 −2.4 0 2.4

(m) Ns = 17, κmax/κmin = 2. (n) Ns = 17, κmax/κmin = 4. (o) Ns = 17, κmax/κmin = 6.

1.5 2.1 0.8 2 2 2

0 0 0 0 0 0

−2 −2 −2 −0.8 −1.5 −2.1 −2 0 2 −2 0 2 −2 0 2 −0.8 0 0.8 −1.5 0 1.5 0 2.1

(p) Ns = 18, κmax/κmin = 2. (q) Ns = 18, κmax/κmin = 4. (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 of the parameters of a monochromatic wavefield from measurements corrupted by additive white Gaussian noise. The approach accounts for all the measurements and all the wavefield parameters jointly. It also allows to model different types of waves and to combine measurements from different types of sensors. The technique can cope with arbitrary sensor positions and different sampling rates in each sensor. This method was presented in Chapter 2 and Chapter 3. The method makes an optimal use of the available measurements according to the ML criterion. All the wavefield parameters and all the measurement are modelled jointly within this estimation framework. We showed that our implementation 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 estimation accuracy given by the Cramér-Rao bound (CRB). The simultaneous presence of multiple waves, of possibly different type, is also addressed. The wave type and number are chosen by the algorithm according to the Bayesian information criterion (BIC). By accounting for multiple waves, the estimates of the parameters of each wave are refined as parameters are iteratively re-estimated. This allows to identify waves with smaller amplitude that would not be visible by modelling a single wave. We showed how to account for different noise variance on each channel, by properly merging the information from sensors with different noise level. The proposed technique estimates the noise variance in each channel. This brings about various advantages. It enables us to use sensors of different type with different noise levels. A misplaced or badly working sensor, may exhibit a higher noise level and will be automatically given less weight in the estimation process. The factor graph approach has been used to define the statistical model and to design an algorithm for the computation of the likelihood of the observations. Thanks to this framework it has been possible to derive sufficient

129 130 statistics enabling us to reduce the computational complexity of the estimation algorithm. We showed how it is possible to extend the method to diﬀerent wave types and measurement types. We showed applications to the analysis of Love waves and Rayleigh waves from recordings of either translational or rotational motions.

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) and on real data from seismic surveys in Switzerland. We showed how this method improves the identification of Rayleigh wave and Love wave dispersion curves when compared with existing methods. We have also shown that modelling multiple waves enables us to detect weaker waves that are not visible with traditional methods. Rayleigh wave ellipticity curve is also retrieved including information about the prograde or retrograde particle motion. This is useful in mode separation and in the identification of singularities of the ellipticity (i.e., peaks and minima of the H/V representation of the ellipticity). We present another example to demonstrate the capabilities of the proposed 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. The array layout is depicted in Fig. 6.1. The array analysed has an aperture of 240 m and records for 2.5 hours. The field measurement are processed with the ML method presented in this thesis. The simultaneous presence of up to three Love or Rayleigh waves is modelled. In Fig. 6.2, the estimated wavefield parameters of Rayleigh waves are presented. The dispersion curve of the fundamental mode and the first higher mode of Rayleigh wave are depicted in Fig. 6.2(a) and in Fig. 6.2(b), respectively. The black dashed curves depict a manual selection of the dispersion curve. The solid red curves are found automatically as the median value of the estimates in the selection. The Rayleigh ellipticity angle for both modes is retrieved over a broad range of 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 is horizontally polarized (V=0). Just above 2 Hz another singularity is found where the Rayleigh wave particle motion changes from prograde to retrograde and the motion is vertically polarized (H=0). The ellipticity of the first higher mode is shown in Fig. 6.2(d). A singularity is found just below 3 Hz, where the Rayleigh particle motion is horizontally polarized (V=0) and the particle motion changes from retrograde to prograde. In Fig. 6.3, the estimated wavenumbers of Love waves are presented. The fundamental mode and the first higher mode of Love wave are depicted in Fig. 6.3(a) and in Fig. 6.3(b), respectively. 131

Figure 6.1: Geometry of the sensor array used in the Yverdon survey. The inlet pinpoints the location of the array within Switzerland. The geographic coordinates are Swiss coordinates (CH1903).

0.03 0.03

0.025 0.025

0.02 0.02

0.015 0.015

0.01 0.01 Wavenumber [1/m] Wavenumber [1/m] 0.005 0.005

0 0 1 2 3 4 5 6 1 2 3 4 5 6 Frequency [Hz] Frequency [Hz] (a) Rayleigh wavenumber, fundamental mode. (b) Rayleigh wavenumber, ﬁrst higher mode. π π 2 2 [rad] [rad] π π ξ ξ 4 4

0 0

- π - π 4 4 Ellipticity angle Ellipticity angle

- π - π 2 1 2 3 4 5 6 2 1 2 3 4 5 6 Frequency [Hz] Frequency [Hz] (c) Rayleigh ellipticity angle, fundamental (d) Rayleigh ellipticity angle, ﬁrst higher mode. mode. Figure 6.2: Estimated Rayleigh wavenumber and ellipticity angle from the Yverdon survey. 132

0.03 0.03

0.025 0.025

0.02 0.02

0.015 0.015

0.01 0.01 Wavenumber [1/m] Wavenumber [1/m] 0.005 0.005

0 0 1 2 3 4 5 6 1 2 3 4 5 6 Frequency [Hz] Frequency [Hz] (a) Love wave fundamental mode. (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 joint processing translational motions and rotational motions measurements. In this application, sensors of different type are used together and is therefore necessary to consider the different noise level. For the single sensor setting, we showed an example of the retrieval of both Love wave and Rayleigh wave dispersion curves. Rayleigh wave ellipticity curve is also retrieved. These quantities were compared with the same quantities obtained from an array of translational sensors. For the array of sensors setting, we showed an application to the processing of recordings from an array of mixed three- and six- components sensors. We found that the joint analysis of translational and rotational measurements greatly improves the retrieved dispersion curve. Analysis of Fisher information (FI) enables us to understand and quantify the sources of uncertainty affecting the accuracy of wavefield parameter estimation. In particular, we derive analytical expressions of CRB for the parameters of interest. A Monte Carlo simulation suggest that, in the considered setup, the MSEE obtained using a single six-components sensor is significantly larger that the MSEE obtained using an array of five sensors.

6.2 Sensor Placement for the Analysis of Seismic Surface Waves

Chapter 5 dealt with the design of array geometries for the analysis of Love waves and Rayleigh waves. We explained in detail how the array geometry affects the MSEE of parameters of Love waves and Rayleigh waves. We distinguished between gross errors, occurring at low SNR and fine errors occurring at high SNR. We showed how the 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 a Rayleigh wave. Relying on the considerations concerning the sources of error, we propose a cost function suitable for the design of the array geometry with particular focus on the estimation of the wavenumber of both Love and Rayleigh waves. The proposed cost function is however difficult to minimize. To circumvent this, we consider a relaxation of the original optimization problem and we propose an algorithm to minimize the relaxed problem. Numerical experiments verify the effectiveness of our cost function and resulting array geometry designs, leading to greatly improved estimation performance in comparison to arbitrary array geometries, both at low and high SNR levels. Moreover, the optimized array geometries obtained with the proposed sensor placement algorithm exhibit a symmetry suitable for the field deployment.

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 alluvial sediments, see e.g., Roten et al. (2006). At the present state, a long recording is split in intervals of determined duration and each interval is processed separately. This approach does not consider that certain waves may have longer duration than other, possibly depending on the nature of their source. We argue that a strategy able to find adaptively the duration of a wave may be both interesting from a scientific point of view and improve the estimation results. Moreover, wavefield parameter from each interval of a long recording are estimated independently. Wavefield parameters from each frequency analysed within the same interval are also estimated independently. This approach ne- glects two facts. First, at a certain frequency the wavenumber of a given wave type and propagation mode is the same during the whole recording. Second, for a given time interval the wavenumber of a certain wave type and propagation mode are strongly correlated at neighbouring frequencies, i.e., the dispersion curves are continuous and typically do not exhibit sudden variations. We believe that these two facts may be exploited to significantly improve the accuracy of the estimated parameters. Concerning the proposed algorithm for sensor placement, we envision two different extensions. First, the presence of physical obstructions in the field should be considered. In fact, in practical applications there are often limitations to where it is possible to position the seismic sensors. Second, possibility of sensor failures should be considered. The robustness of the proposed sensor placement algorithm should be evaluate and, if necessary, a suitable strategy to address this problem should be developed. The algorithms developed in this thesis also need to be implemented as standalone software packages and distributed to interested parties within the Swiss Seismological Service. In this way, the methods proposed in this the- 134 sis will contribute to the site-characterization tasks performed at the Swiss Seismological Service. Appendix A

Seismic Waves Estimation and Waveﬁeld Decomposition with Factor Graphs

Stefano Maranò1,Christoph Reller2, Donat Fäh1, and Hans-Andrea Loeliger2

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 number of measurement at the same location (e.g. direction, polarization, translation, and rotation). One question arising naturally is how to properly process such large and diverse information, possibly from sensors of different kinds. In this paper we propose a technique for the analysis of vector wavefields and show an application to the seismic wavefield. The contributions of this paper are the following: i) We provide a framework to perform maximum likelihood parameter estimation of any wave type, modeling jointly all the measurements and parameters; ii) In the same framework, we address wave superposition by gradually decomposing the wavefield; iii) We also propose an iterative algorithm for noise variance estimation.

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 with sensors 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 the contributions of different waves. In the analysis of physical wavefields we are typically interested in studying vector fields of the form u(p, t): R4 RC. The quantity u RC depends on the position p R3, time t, and→ wavefield parameters θ.∈ The value of C depends on the∈ sensor we use to measure the wavefield. The wavefield is measured by means of an array of N sensors. We call L = NC the overall number of channels. In presence of multiple sources, M waves coexist at the same time in the wavefield. Assuming a linear medium, the superposition of these waves is measured in each channel. We now restrict our analysis to a monochromatic wavefield with known angular frequency ω. In this setting, we model the signal on each channel as the sum of scaled and delayed copies of underlying reference sinusoids

M (ℓ) (m) (m) (m) (m) u (p, t) = ρℓ ρ0 cos ωt + φℓ + φ0 . (A.1) m=1 X With this signal model, the m-th wave is parametrized by the amplitudes and , (m) (m) (m) phases ηm ρℓ , φℓ ℓ=0,...,L. The m-th reference sinusoid ρ0 cos(ωt + (m) { (m) } (m) φ0 ) is defined by ρ0 and φ0 . However, our interest lies in the estimation of wavefield parameters θm governing the amplitudes and the phases measured at the different sensors. The parameters η and θ are related by means of a mapping Γ: θ η. In general, ρℓ, φℓ ℓ=1,...,L can be influenced by different quantities such→ as velocity and direction{ } of propagation (i.e., by the wave vector), wave polarization, wave attenuation, sensor directional gain, instrument response, and others. Each signal is measured as

(ℓ) (ℓ) (ℓ) Yk = u (p, tk) + Zk (A.2) at times t and is corrupted by Gaussian noise Z(ℓ) iid 0, σ2 . k k ∼ N ℓ A.2 Seismic Waveﬁeld

The seismic wavefield (i.e., elastic waves propagating through the earth) offers an 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) seismometers on the surface of the earth. Each sensor measures the ground velocity along the direction of the axes of the coordinate system x, y, and z. For the sake of simplicity, we provide wave equations of the displacement field u, despite the ∂u actual measurement is the velocity field ∂t .The displacement can be described by the vector field

u(p, t) , (u (p, t), u (p, t), u (p, t)) : R4 R3 . x y z → We restrict our interest to small aperture arrays and work with a ﬂat earth model. We use a three-dimensional, right-handed Cartesian coordinate system with the z axis pointing upward. The azimuth ψ is measured counterclockwise from the x axis. 137

In this paper, we study waves propagating near the surface of the earth and having a direction of propagation lying on the horizontal plane z = 0. The wavefield is composed of the superposition of several Rayleigh and Love waves. The wave equations we describe hereafter are valid for z = 0 and for plane wave fronts. Rayleigh waves exhibit an elliptical particle motion confined in the vertical plane perpendicular to the earth’s surface and defined by the direction of propagation of the wave. The particle displacement generated by a single Rayleigh wave at position and time (p, t) is

u (p, t) = α sin ξ cos ψ cos(ωt κTp + ϕ) x − uy(p, t) = α sin ξ sin ψ cos(ωt κTp + ϕ) (A.3) − π u (p, t) = α cos ξ cos(ωt κTp + + ϕ) . z − 2

The direction of propagation of a wave is given by the wave vector κ , κ (cos ψ, sin ψ, 0) T, whose magnitude κ is the wavenumber. The quantity tan ξ is called ellipticity of the Rayleigh wave and determines the eccentricity and the sense of rotation of the particle motion. Love waves exhibit a particle motion conﬁned on the horizontal plane, the particle oscillates transversely with respect to the direction of propagation. The particle displacement generated by a single Love wave is

u (p, t) = α sin ψ cos(ωt κTp + ϕ) x − − u (p, t) = α cos ψ cos(ωt κTp + ϕ) (A.4) y − uz(p, t) = 0 .

With these wave equations in mind, we can now give an explicit expression for 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 , φ = (sin ξ cos ψ, κTp ) (n,1) (n,1) − n ρ , φ = (sin ξ sin ψ, κTp ) (n,2) (n,2) − n π ρ , φ = cos ξ, κTp + (n,3) (n,3) − n 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 , φ = ( sin ψ, κTp ) (n,1) (n,1) − − n ρ , φ = (cos ψ, κTp ) (n,2) (n,2) − n 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-th channel. 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

L K 2 1 − y(ℓ)−u(ℓ) /2σ2 p(y η) = e k k ℓ , (A.5) 2 | 2πσℓ ℓY=1 kY=1 where we rely on K discrete-timep observations for each channel and deﬁne , (ℓ) k=1,...,K (ℓ) , (ℓ) y yk ℓ=1,...L and uk u (pℓ, tk). Instead{ } of computing (A.5), we model it by means of a factor graph Loeliger (ℓ) et al. (2007). For every signal Yk , we consider a second-order state space model with state X(ℓ) R2 k ∈ (ℓ) A (ℓ) Xk−1 = kXk (ℓ) (ℓ) (ℓ) Yk = CXk + Zk , where A , rotm(ω(t t )) is a clockwise rotation matrix rotm(β) , k k−1 − k cos β − sin β , and the measurement matrix C , (0, ω) accounts for the sin β cos β − ∂u derivative ∂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 the following L equations

M M (ℓ) H(m) (m) XK = ℓ um = Sℓ , (A.6) m=1 m=1 X X H(m) , (m) (m) (m) , H(m) where ℓ ρℓ rotm(φℓ ) are the constraint matrices, Sℓ ℓ um is the contribution on the ℓ-th channel of the m-th wave, and the state vector of , (m) (m) (m) the m-th reference sinusoid is um ρ0 (cos(ωtK + φ0 ), sin(ωtK + φ0 ))T . 6m The corresponding factor graph is shown in Fig. A.1(b), where Sℓ represents the contribution on the ℓ-th channel of all but the m-th wave. The overall graph is shown in Fig. A.1(c). Using the sum-product algorithm on the factor graph we can compute the likelihood function p(y η). We use Gaussian messages parametrized by mean vector, covariance matrix,| and scale factor. We provide a detailed description in Reller et al. (2011).

Parameter Estimation We now focus on the estimation of waveﬁeld parameters θ, in the case of a single wave (M = 1) and known noise variance. We introduce the set of all the parameter mappings of interest. The maximum likelihood (ML)G estimate of waveﬁeld parameter is given by

(Γˆ, θˆ) argmax p y Γ(θ) . (A.7) ∈ Γ∈G,θ∈dom Γ This maximization allows us to ﬁnd the most likely wave type with the most likely parameters. 139

X(ℓ) X(ℓ) S(m) X(1) k A k−1 = H(m) 1 K = k 1 + (6m) S1 C

(ℓ) (m) (L) Zk SL XK + H(m) + N L (ℓ) (ℓ) (6m) um Yk = yk SL

(a) State space model. (b) Glue factor g.

(1) (1) (1) (1) XK Xk Xk−1 X1

(1) (1) (1) (1) Yk = yk Y1 = y1

(L) (L) (L) (L) XK Xk Xk−1 X1

g (L) (L) (L) (L) Yk = yk Y1 = y1

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 of the likelihood function might be impractical. We propose to gradually increase the number of waves modeled by the graph and perform smaller maximizations on the wavefield parameters of each wave ˆ6m ˆ6m separately. In practice, once we insert an estimate sℓ of Sℓ , we perform the maximization over θm as in (A.7). Each maximization increases the likelihood and convergence is guaranteed.

Noise Variance Estimation The ML noise variance estimate is given by

K 2 2 1 (ℓ) → σˆℓ = yk C rotm (ω(tk tK )) −mX(ℓ) , K − − K Xk=1 → M H(m) → → 2 with m (ℓ) = m=1 ℓ mU m . Since the messages mU m depend on σˆℓ , this − XK − − leads to an iterative algorithm where noise variance and messages are estimated P iteratively. 140

0.9

0.8

0.7

0.6

0.5

0.4 Amplitude [m] αˆ(R) 0.3 1 (R) αˆ2 0.2 (L) αˆ3 0.1 (L) αˆ4 0 2 4 6 8 10 12 14 16 18 Iteration

Figure A.2: Estimated amplitude at diﬀerent iterations. The graph accounts for 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 Rayleigh and two Love waves. The number of waves M = 4 is known. Waves are monochromatic at known frequency of 1 Hz. We use an array of 14 triaxial sensors, 500 samples, and 5 seconds of observation. Measurements are corrupted by additive white Gaussian noise, with different variance in each channel. We look for both Rayleigh and Love waves, i.e., , Γ(R), Γ(L) . True wave- (R) π π G (R{) π} π π field parameters are θ1 = (0.9, 0, 0.03, 4 , 4 )T, θ2 = (0.8, 4 , 0.03, 2 , 4 )T, (L) π π (L) θ3 = (0.7, 3 , 0.04, 4 )T, and θ4 = (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 factor graph is enlarged to account for additional waves at iterations 6, 10, and 15 as the likelihood (not shown) converges to a stable value. Similarly, Fig. A.3 2 shows the the estimates of noise variance σℓ . Sudden decrease in estimated variance in the graph correspond to the inclusion of an additional wave in the graph. Fig. A.4 depicts log p y Γ(L)(θ(L)) , as a function of wavenumber κ and azimuth ψ in polar coordinates (κ, ψ). In Fig. A.4(a) it is possible to see at π (L) ψ3 = 4 one stronger peak associated with the wave parametrized by θ3 and no− other strong peaks are visible. At iteration 14, only one Love wave (L) remains in the wavefield (the wave parametrized by θ4 ) and the associated peak, located at ψ4 = π, is now clearly visible, as shown in Fig. A.4(b). At the last iteration, no more waves remain in the the residual wavefield, Fig. A.4(c). We now use a more sophisticated synthetic wavefield developed in the SESAME project Bard, P.-Y. (2008); Bonnefoy-Claudet et al. (2006a). This synthetic dataset captures the complexity of the seismic wavefield, accounting for the simultaneous presence of several seismic sources, emitting both short burst of energy and longer harmonic excitations. It is a wavefield of ambient vibrations, where the wavefield is dominated by surface waves (i.e., Rayleigh 141

0.9 ]

2 0.8

0.7

0.6 σˆ2 0.5 1 σˆ2 0.4 2 σˆ2 Noise Variance0.3 [m 3 2 σˆ4 0.2 2 σˆ5 0.1 2 σˆ6 0 2 4 6 8 10 12 14 16 18 Iteration

Figure A.3: Estimated noise variance at diﬀerent iterations. Only six channels are shown.

0.06 0.06 0.06 ] ] ] 1 1 1 − − −

[m 0 [m 0 [m 0 ψ ψ ψ sin sin sin κ κ κ −0.06 −0.06 −0.06 −0.06 0 0.06 −0.06 0 0.06 −0.06 0 0.06 κmcos ψ [ −1] κmcos ψ [ −1] κmcos ψ [ −1] (a) Iteration 1 (b) Iteration 14 (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 standing waves). We use an earth model of a layer with low seismic velocities over a half-space with higher velocities. We use 38 sensors and solely 10 seconds of recording. Diﬀerent frequencies are processed independently. Of practical interest is the phase velocity dispersion of surface waves Aki & Richards (1980). We deﬁne , Γ(R) and initially model a single Rayleigh wave, i.e., M = 1. In Fig.G A.5,{ the estimates} of the wavenumbers κ (black dots) suggest the Rayleigh wave dispersion curves. For comparison, the theoretical dispersion curves are depicted by lines. In Fig. A.6, the number of waves modeled is increased to M = 3. It is shown that increasing the number of waves modeled by the factor graph, allows to better retrieve the fundamental and the higher modes.

A.5 Conclusions

We have developed a technique to perform ML estimation of wavefield parameter of any wave type. The technique accounts for different noise variance on each channel, by properly merging the information from sensors with different noise level. In the same framework, we address the superposition of multiple 142

0.07 Fundamental mode 0.06 ]

1 First higher mode − 0.05 Second higher mode Est. wavenumber κˆ 0.04

0.03 Wavenumber [m 0.02

0.01

0 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz]

Figure A.5: Rayleigh wave dispersion curve, M = 1.

0.07 Fundamental mode 0.06 ]

1 First higher mode − 0.05 Second higher mode Est. wavenumber κˆ 0.04

0.03 Wavenumber [m 0.02

0.01

0 2 3 4 5 6 7 8 9 10 11 12 Frequency [Hz]

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 technique 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 function for the amplitudes and phase shifts of noisily observed phase-locked and amplitude-constrained sinusoids. The sinusoids are assumed to be coupled based on a set of parameters as, e.g., measurements of a monochromatic wave field. A factor graph is used to formulate the probability density function of the observations given the parameters. The factor graph consists of one second-order state-space model per signal and one additional factor connecting all the final states. Because the parameters appear only in this latter factor, we are able to formulate a sufficient statistic for parameter estimation and signal detection in terms of messages in the factor graph. In special cases, the general form of the sufficient statistic reduces to the discrete Fourier transform. As extensions we provide iterative algorithms for approximate maximum likelihood estimation of the noise variances and the parameters of superposed waves.

B.1 Introduction

Phase-coupled and amplitude-constrained sinusoids play an important role in many ﬁelds ranging from sensor arrays in seismology (Maranò et al., 2011), acoustics, and electromagnetics to multi-terminal communication. Parameter estimation for, and detection of, uncoupled sinusoids is described e.g. in (Kay,

143 144

(1) (1) (1) (1) XK Xk+1 Xk X1 , S1 (1) (1) yk y0

(L) (L) (L) (L) XK Xk+1 Xk X1 , S gθ L (L) (L) yk y0

Figure B.1: Overall factor graph

(ℓ) (ℓ) Xk+1 X A = k U = u(θ) S H 1 C = 1(θ)

Z(ℓ) 2 k σ (0, ℓ ) + N SL (ℓ) (ℓ) HL(θ) Yk = yk

Figure B.3: Glue factor gθ Figure B.2: State-space model for sinusoid

1993) and (Kay, 1998). However, as soon as we consider some coupling, a linear model does not apply anymore. This paper provides a uniﬁed and general approach to estimation and detection of coupled sinusoids based on a factor graph representation (Loeliger et al., 2007). Consider L discrete-time sinusoidal signals

(ℓ) ξk = αℓ cos(Ωk + ψℓ) , (B.1) where ℓ = 1,...,L enumerates the signals and k = 0,...,K 1 is the time index. All L signals have the same, known frequency Ω but differ− in amplitude (ℓ) (ℓ) (ℓ) (ℓ) αℓ and phase ψℓ. We observe the noisy signal Yk = ξk + Zk , where Zk 2 are zero-mean white Gaussian noises with noise variances σℓ for ℓ = 1,...,L. 2 Note that we allow σℓ 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 that the amplitudes αℓ and phase shifts ψℓ are constrained by some parameter vector θ via some mapping

Γ: θ (α , ψ ),..., (α , ψ ) . (B.2) 7→ 1 1 L L As a toy example assume that we have two signals (B.1) with unconstrained amplitudes α1 and α2 but with the same phase ψ , ψ1 = ψ2. We are interested in α1, α2 and ψ. In this example one possible choice is θ = (ρ0, φ0, β) and Γ(θ) = (ρ0, φ0), (βρ0, φ0) . 145

For a more relevant example consider the estimation of seismic wavefields measured by a sensor array. In this example, θ may contain wavefield parameters such as wave type, velocity of propagation, angle of arrival, etc. The mapping Γ would include sensor characteristics and positions. (This setting is treated in more detail in (Maranò et al., 2011).) We use the term wave for ξ , ξ(ℓ) induced by the { k }ℓ=1,...,L,k=0,...,K−1 parameters θ given the mapping (B.2). For the noisy signal we define Y (ℓ) , (ℓ) (ℓ) , (1) (L) Y0 ,...,YK−1 and Y Y ,..., Y . Given observations Y = y, we want to formulate the likelihood function f(y θ) in order to make ML estimates | θˆ = argmax f(y θ) (B.3) θ | and to compute the generalized log-likelihood ratio (LLR)

f y θˆ ln , (B.4) y f( 0) |H where the hypothesis 0 is the presence of only noise. (The LLR is usually used for signal detectionH (Kay, 1998).) Clearly, the likelihood can be written as

L K−1 2 1 − y(ℓ)−ξ(ℓ)(θ) 2σ2 f(y θ) = e k k ( ℓ ) . (B.5) 2 | 2πσℓ ℓY=1 kY=0 In Section B.2 we show that f(ypθ) is a function of the quantities Hℓ(θ) ← ← | ∈ R2×2 W 2 R2×2 (ℓ) R , −Sℓ (σℓ ) , and m−Sℓ (y ) for ℓ = 1,...,L. The complexity ←∈ ← ∈ ← ← for computing W− and m is linear in K and for fixed H , W− , m the Sℓ −Sℓ ℓ Sℓ ←−Sℓ complexity for computing this function is linear in L. Note that W− and ← Sℓ m−Sℓ do not depend on θ and form a sufficient statistic. In Section B.3 we show how this sufficient statistic simplifies in restricted settings and we show a connection with the discrete Fourier transform (DFT). In Section B.4 an iterative algorithm for approximating the ML estimate of the noise variances in each signal is given. Finally, in Section B.5 we formulate an iterative algorithm to 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 with state vector Xk as (ℓ) A (ℓ) Xk = Xk+1 , (B.6) (ℓ) (ℓ) (ℓ) Yk = CXk + Zk , (B.7) where A is a rotation matrix A , rotm( Ω) , (B.8) − cos α sin α rotm α , − , (B.9) sin α cos α ! 146 and C , (1, 0). The corresponding (Forney) factor graph is depicted in Fig. B.2. Note that the time progression in Figs. B.2 and B.1 is from right , (ℓ) to left. We use the abbreviation Sℓ XK for the ﬁnal system state.

Glue Factor Without loss of generality we can assume that θ contains the overall amplitude ρ0 = 0 and overall phase φ0 of an unknown reference sinusoid ρ0 cos(Ωk + φ0). By6 letting

cos(ΩK + φ0) u(θ) , ρ0 (B.10) sin(ΩK + φ0)! be the state of this reference sinusoid at time K we can express the coupling between the signals as Sℓ = Hℓ(θ) u(θ) , (B.11) where H (θ) , ρ rotm(φ ), ρ , α /ρ , and φ , ψ φ . ℓ ℓ ℓ ℓ ℓ 0 ℓ ℓ − 0 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)

L gθ(s ,..., s ) = δ s H (θ) u(θ) , (B.12) 1 L ℓ − ℓ ℓ=1 Y 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 and represents the probability density function

f(y, x θ) = f y, x u(θ), H(θ) , (B.13) | , (ℓ) H , H T H T T with X Xk ℓ=1,...,L,k=1,...,K and (θ ) 1(θ) ,..., L(θ) . Instead{ of using} the brute force calculation (B.5), we use sum-product message passing (Loeliger et al., 2007) in the factor graph to compute the likelihood → function. We use arrows to distinguish between forward messages ( ) in the ← −· same direction as the edge and backward messages (−) in the opposite direction. · By marginalization and due to the deﬁnition of the sum-product rule we can write ← f(y θ) = f(y, x θ) dx = µ (u(θ)) . (B.14) | | −U Z Since the state-space models (Fig. B.2) do not depend on θ, we can compute ← the 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 sufficient statistic. All messages in the factor graph at hand are (potentially scaled and degenerate) multivariate Gaussian probability density functions. In this paper we prefer to write a message (e.g. for an edge X in forward direction) in the form → → → → TW− TW− −→ , −x X x/2+x X mX −µX (x) −γX e , (B.15) → → V W−1 → where −X = − X 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 uˆ = m−U . We set u(θ) = uˆ in (B.14) and get the partially maximized likelihood function from message update rules for Gaussian messages (Loeliger et al., 2007) as 1← ← ← ← ln f y u(θ) = uˆ, H(θ) = mT W− m + ln γ , (B.17) 2 −U U −U −U where ← L ← W H T W H −U = ℓ(θ) −Sℓ ℓ(θ) , (B.18) Xℓ=1 ← L ← W ← H T W ← −U m−U = ℓ(θ) −Sℓ m−Sℓ , (B.19) Xℓ=1 ← L ← ln −γU = ln −γSℓ . (B.20) Xℓ=1 The remaining maximization over Hℓ(θ) is in general non-convex and depends largely on the mapping Γ. We do not treated it here. For the noise hypothesis 0 we constrain X to be zero by setting u = 0 in (B.12). From (B.14) and (B.16)H we get ← ln f(y ) = ln γ , (B.21) |H0 −U so that the partially maximized LLR is f(y u(θ) = uˆ, H(θ)) 1← ← ← LLR(θ) , ln | = mT W− m . (B.22) f(y ) 2 −U U −U |H0 Note that (B.20) does not depend on θ and therefore can be neglected when ← ← θˆ m W finding ML estimates . The sufficient statistic thus consists of −Sℓ , −Sℓ ℓ=1,...,L. It is easy to generalize the state-space models to non-uniform sampling. For A A(ℓ) , (ℓ) this we substitute in (B.6) by time-varying matrices k rotm (tk−1 (ℓ) (ℓ) − tk ) ω , where ω is the continuous time frequency and tk are time stamps (ℓ) of yk . In the same fashion we can easily accommodate time varying noise variances. The likelihood function and the LLR can still be computed by message 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 analytic ← solution for the messages −µSℓ can be proven. ← W K I sin(ΩK) R −Sℓ = 2 2 + 2 rotm(Ω) , (B.23) 2σℓ 2σℓ sin Ω ← K−1 cos(kΩ) W ← 1 R (ℓ) −Sℓ m−Sℓ = 2 yk , (B.24) σℓ sin(kΩ)! kX=0 R , 1 0 where rotm(ΩK) 0 −1 . When viewing the sum in (B.24) as a function of Ω we recognize the real and imaginary parts of the ﬁnite-length discrete-time (ℓ) (ℓ) Fourier transform of y0 , . . . , yK−1 . If we further restrict the frequency Ω to be Ωn , 2πn/K for n = 0,...,K R 1 0 − 1, then = 0 −1 and the expressions above simplify to ← W K I −Sℓ = 2 2 , (B.25) 2σℓ K−1 ← 2 (ℓ) cos(2πkn/K) m−Sℓ = yk . (B.26) K sin(2πkn/K)! kX=0 − (ℓ) (ℓ) The latter consists of the real and the imaginary parts of the DFT of y0 , . . . , yK−1 , scaled by 2/K. If, in addition, the noise variances are the same in all signals, 2 2 i.e., if σℓ = σ for ℓ = 1,...,L, then (B.22) simpliﬁes to

˘T H H T ˘ yn (θ) (θ) yn LLRn(θ) = , (B.27) 2 L 2 Kσ ℓ=1 ρℓ

← ← T where y˘ , K mT ,. .., mT containsP the n’th component of the DFTs n 2 −S1 −SL of the signals. Equation (B.27) is a standard beam-forming result. (See e.g. (Krim & Viberg, 1996).)

B.4 Noise Variance Estimation

, 2 2 In this section we consider the case where the noise variances η σ1 , . . . , σL are not given a-priori but are estimated in an ML sense for both the “signal present” ( ) and the “noise present” ( ) hypothesis. H1 H0 The joint maximization for 1 over (θ, η) is non-convex. We propose to use cyclic maximization (StoicaH & Selen, 2004) by alternating

θˆ = argmax f y θ, ηˆ , (B.28) θ ˆ ηˆ = argmax f y θ, η . (B.29) η Since the likelihood in every iteration cannot decrease, cyclic maximization algorithms are guaranteed to converge. The maximization (B.28) is the same as (B.3) and hence the procedure in Section B.2 applies. To start the algorithm we propose an initial estimate ηˆ 149 based on assuming that the signals are decoupled, i.e., the glue factor (B.12) is replaced by L ← g (s ,..., s ) = δ(s m ) . (B.30) θ 1 L ℓ − −sℓ ℓY=1 With (B.30), these initial ML noise variance estimates are

K−1 1 ← 2 σˆ2 = y(ℓ) CAK−km . (B.31) ℓ K k − −Sℓ Xk=0 ˆ → ˆ Once we have an estimate θ we can calculate −mSℓ by apply θ in the glue factor (B.12) and get the coupled ML noise variance estimates as

K−1 1 → 2 σˆ2 = y(ℓ) CAK−km . (B.32) ℓ K k − − Sℓ Xk=0 2 If the signals are long we might want to avoid the direct computation of σˆℓ . We can approximate (B.31) and (B.32) by ← ← 2 2 mT m σˆℓ ζℓ −Sℓ −Sℓ /2 , (B.33) ≈ − → → → ← σˆ2 ζ2 + mT m /2 mT m (B.34) ℓ ≈ ℓ − Sℓ − Sℓ − − Sℓ −Sℓ 2 respectively, where ζ2 , 1 K−1 y(ℓ) are the signal powers. Using these ℓ K k=0 k ← ← W 2 approximations, the only input to the algorithm is m−Sℓ , −Sℓ , ζℓ ℓ=1,...,L. It ← P {← } 2 W can be shown that m−Sℓ does not depend on σℓ and −Sℓ depends linearly on 2 σℓ . Under the noise hypothesis 0 the ML estimate of the noise variances are ˆ(0) , 2 2 H η (ζ1 , . . . , ζL). The generalized LLR can easily be calculated from (B.5) as (Kay, 1998) ˆ L 2 f y θ, ηˆ, 1 K ζ ln H = ln ℓ . (B.35) (0) 2 σˆ2 f y ηˆ , 0 ℓ=1 ℓ H X

B.5 Extension to Wave Superposition

Assume that we observe a linear superposition of M waves ξ(m) with same frequency Ω, parameters θ , and mappings Γ : θ α(m), ψ(m) ,..., α(m), ψ(m) m m m 7→ 1 1 L L for m = 1,...,M. Our signal model now is M (ℓ) (m) (m) (ℓ) Yk = αℓ cos Ωk + ψℓ + Zk . (B.36) m=1 X

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) , T T T Hℓ(θ1),..., Hℓ(θM ) and state vectors u˜(θ) , u(θ1) ,..., u(θM ) in (B.11) and (B.12). However, the space over which to maximize in (B.17) increases approximately M fold. 150

As an alternative we propose an iterative algorithm based on cyclic maximization (Stoica & Selen, 2004). Assume that we have an estimate θˆ. We pick some m 1,...,M and update the estimate of θ while fixing θ = ∈ { } m { j θˆ . This leads to the following glue factor j }j∈{1,...,M}\{m} L 6m gθ (s ,..., s ) = δ s sˆ H (θ ) u(θ ) , (B.37) m 1 L ℓ − ℓ − ℓ m m ℓ=1 Y ˆ6m , H ˆ ˆ where sℓ j∈{1,...,M}\{m} ℓ(θj ) u(θj ) is the estimated state due to all the waves except for the m-th. The corresponding θˆ can again be found by P ← m maximizing (B.17) where −µU is calculated using the glue factor (B.37). To apply this algorithm we propose the following greedy-type procedure. ˆ Initially, set M = 1 and use the glue factor (B.12) to find θ1. Then repeatedly do the following. Increase M, use the glue factor (B.37) with m = M to ˆ ˆ find θM , and iterate finding θm for m 1,...,M until convergence. This algorithm 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 this is the DFT) for the ML estimation of wave parameters. The suﬃcient statistic can be used to devise iterative algorithms for the estimation of superposed waves and of the noise variances. Appendix C

Estimation of Waveﬁeld Parameters of a Single PWave at the Free Surface

C.1 Introduction

We are interested in the estimation of body waves from array recordings of ambient vibrations. We consider the case of a single incident P wave. At the free surface, where the array is positioned, a body wave will exhibit an apparent velocity larger than the true velocity of propagation. The apparent velocity depends on the velocity of the wave in the uppermost layer and the incidence angle. Assuming an homogeneous first layer with thickness much larger than the wavelength, the ground motion observed at the surface is explained by considering the superposition of the incident P wave together with the reflected P wave and S wave. In the following sections, we describe the wave equations used to model the ground displacement, the approach for wavefield parameters estimation, and show some numerical results.

C.2 System Model

Our interest lies in the estimation of P wave parameters from the analysis of the ground displacement induced by a single incident P wave. The ground displacement is measured by a planar array of Ns sensors positioned at the surface of the earth. In such setting, it is necessary to consider the P-SV coupling at the free surface. We model the wavefield u measured at the surface as the sum of the three wavefields 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 wavenumbers 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 and wavevector κ1 = κP(cos ψ sin η, sin ψ sin η, cos η) is

151 152

u(1)(p, t) = α cos ψ sin η cos(ωt κ p + ϕ) (C.1) x − 1 · u(1)(p, t) = α sin ψ sin η cos(ωt κ p + ϕ) (C.2) y − 1 · u(1)(p, t) = α cos η cos(ωt κ p + ϕ) . (C.3) z − 1 · 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

4 sin2 η cos η γ sin2 η (γ 2 sin2 η)2 R = − − − , (C.4) PP 2 2 2 4 sin η cos ηpγ sin η + (γ 2 sin η)2 − − 2 2 p where γ = vP/vS. We observe that γ > 2. The displacement at the free surface due to the reﬂected P wave is given by u(2)(p, t) = R α cos ψ sin η cos(ωt κ p + ϕ) (C.5) x PP − 2 · u(2)(p, t) = R α sin ψ sin η cos(ωt κ p + ϕ) (C.6) y PP − 2 · u(2)(p, t) = R α cos η cos(ωt κ p + ϕ) , (C.7) z − PP − 2 · where κ2 = κP(cos ψ sin η, sin ψ sin η, cos η). Using Snell’s law the angle η′ of the− reﬂected SV wave is found

sin η sin η′ = . (C.8) vP vS

The reflection coefficient RPS for the SV wave is 2 κS 4 sin η cos η(γ 2 sin η) RPS = − − . (C.9) κP · 4 sin2 η cos η γ 2 sin2 η + (γ 2 sin2 η)2 − − The ground displacement due top the reflected SV wave is

u(3)(p, t) = R α cos ψ cos η′ cos(ωt κ p + ϕ) (C.10) x − PS − 3 · u(3)(p, t) = R α sin ψ cos η′ cos(ωt κ p + ϕ) (C.11) y − PS − 3 · u(3)(p, t) = R α sin η′ cos(ωt κ p + ϕ) , (C.12) z − PS − 3 · where κ = κ (cos ψ sin η, sin ψ sin η, κ 2/κ2 sin2 η). 3 P − S P − With the assumption of a linear medium,q the total wavefield observed at the 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 can replace κ1, κ2, and κ3 with κ˜ = κP(cos ψ sin η, sin ψ sin η, 0) and obtain u (p, t) = α cos ψ((1 + R ) sin η R cos η′) cos(ωt κ˜ p + ϕ) x PP − PS − · u (p, t) = α sin ψ((1 + R ) sin η R cos η′) cos(ωt κ˜ p + ϕ) (C.13) y PP − PS − · u (p, t) = α((1 R ) cos η R sin η′) cos(ωt κ˜ p + ϕ) , z − PP − PS − · 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 that the 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 perform ML estimation of wavefield parameters. The wavefield parameter vector (P) for the considered scenario is θ = (α, ϕ, κP, ψ, η, κS), cf. (C.13). We extend the ML method proposed in Maranò et al. (2011) to model an incident P wave. We remark that the P wavenumber κP and the S wavenumber κS that are estimated are the actual wavenumbers in the first layer and are not related to the apparent velocity of the incident P wave. In addition, the inclination angle η corresponds to the incidence angle of the P wave and not to the oscillation angle observed at the surface.

C.4 Numerical Results

Monochromatic Wavefield We simulate the ground displacement induced by a single monochromatic incoming P wave. The ground displacement accounts for the P-SV coupling described in the following section. In particular, we consider an incident P wave with azimuth ψ = π/5, and different incidence angle (or inclination angle). The body wave velocity in the uppermost layer are vP = 500 m/s and vS = 200 m/s. Therefore, considering a monochromatic wave at 5 Hz, the true values for P wave wavenumber and S wave wavenumber are κP = 0.01 1/m and κS = 0.025 1/m, respectively. Parameters are estimated using the proposed ML method. First, we consider a P wave with a shallow incidence angle of 80◦. In Fig. C.1, two slices of the LLF are shown. In Fig.C.1(a) the LLF as a function of κP and κS is depicted. In Fig.C.1(b) the LLF as a function of ψ and η is depicted. The true maximum is marked with a white cross. In both figures it is 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 the LLF around the maximum value. The maximum value, pinpointed by a white cross, is correctly found and the ML estimate of the parameters matches with the true values. Second, we also consider a steeper incidence angle corresponding to the critical angle between the first layer and the half-space. We let the P wave velocity in the half-space be 2000 m/s. The critical angle is then

500 η = arcsin 14.5◦ . (C.14) c 2000 ≈ In Fig. C.2, slices of the LLF are depicted. Similarly to the previous example, the maximum of the LLF identifies the true wavefield parameters. We showed that, under the good conditions of large SNR the retrieval of wave parameters from a single incident P wave is possible. From the shape of the LLF we draw the following considerations on estimation 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

0.035 0

0.03 [rad]

0.025 η [1/m]

P 0.02 κ + π 4 0.015 Inclination 0.01

0.005 P wavenumber + π 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 0 π/2 π 3π/2 2π S wavenumber κS [1/m] Azimuth ψ [rad]

(a) Slice of the LLF as a function of κP and (b) Slice of the LLF as a function of Azimuth κS. and Inclination. Figure C.1: Slices of the LLF for the estimation of waveﬁeld parameters of an incoming P wave. Shallow incidence angle (η = 80◦).

0.035 0

0.03 [rad]

0.025 η [1/m]

P 0.02 κ + π 4 0.015 Inclination 0.01

0.005 P wavenumber + π 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 2 0 π/2 π 3π/2 2π S wavenumber κS [1/m] Azimuth ψ [rad]

(a) Slice of the LLF as a function of κP and (b) Slice of the LLF as a function of Azimuth κS. and Inclination. Figure C.2: Slices of the LLF for the estimation of wavefield parameters of an incoming P wave. Critical incidence angle (η = 14.5◦). maxima. It can be verified by testing the non-singularity of the Fisher information matrix (FIM). Secondly, the model carries very little information about vS. Intuitively, this can be understood by the little change in the value of likelihood for different κP, cf. Fig. C.1(a) and Fig. C.2(a). This could also be verified by inspection of the FIM. The reason for this is that vS only affects the angle of oscillation at the surface, which is difficult to estimated accurately. As a consequence, we expect 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 the SESAME project (Bard, P.-Y., 2008). The structural model is a layer over a half-space and it is depicted in Fig. C.3(a). The P wave and S wave velocities 155

0 P wave S wave 60 −5

−10 40

−15 y [m] Depth [m]

−20 20

−25

−30 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 −40 −20 0 20 Velocity [m/s] x [m] (a) Velocity proﬁle. (b) Array layout.

1100 1100

1000 1000

900 900

800 800

700 700

600 600

500 500

Velocity [m/s] 400 Velocity [m/s] 400

300 300

200 200

100 100

0 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Frequency [Hz] Frequency [Hz] (c) Rayleigh phase velocity for diﬀerent (d) Rayleigh group velocity for diﬀerent modes. modes. 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 wave and 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 parameter estimation. The recording is split in windows of duration 0.5 seconds. Phase velocity of the different modes of surface waves can be computed from the structural model using the the propagator matrix method (Gilbert & Backus, 1966). Phase velocity of several modes of Rayleigh wave is shown in Fig. C.3(c). The Rayleigh wave group velocity is shown in Fig. C.3(d). At any angular frequency ω, the group velocity vg is related to the phase velocity v as dω dv v = = v + κ . (C.15) g dκ dκ In Fig. C.4, the wavenumber and the ellipticity angle estimated for Rayleigh waves are shown. The estimated parameters are depicted in gray and black. The lines represent the theoretical values of the parameters for the different modes. The fundamental mode and part of the first higher mode of the Rayleigh wave are correctly estimated as shown in Fig. C.4(a). In Fig. C.4(b), it is possible to follow the ellipticity angle of the Rayleigh wave especially for the 156

0.1 π 2

0.09

0.08 [rad] π ξ 0.07 4

0.06

0.05 0

0.04

0.03

Wavenumber [1/m] π -Ellipticity angle 4 0.02

0.01

0 - π 2 4 6 8 10 12 14 16 18 20 2 2 4 6 8 10 12 14 16 18 20 Frequency [Hz] Frequency [Hz] (a) Rayleigh wave wavenumber. (b) Rayleigh wave ellipticity angle.

Figure C.4: Summary of the estimated Rayleigh wave parameters for the M2.1 dataset. fundamental mode. In both ﬁgures, there are no estimates above 14 Hz due to limitations 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 and S wave is also depicted as a straight line (points of constant velocity). Equivalently, the estimated P wave and S wave velocities are plotted in in Fig. C.5(c) and Fig. C.5(d). A line depicts the theoretical velocity for the body waves. Both in the wavenumber and in the velocity representation, the estimated values do not resemble the expected values. The estimates of the incidence angle and of the azimuth angle for the P waves are shown in Fig. C.5(e) and Fig. C.5(f), respectively. Very shallow incidence angles and vertical incidence angles have been removed from all the results shown. The resonance frequency of a P wave in the layer is f0 = vP/4h = 5 Hz, where h = 25 m is the thickness of the layer. Therefore it is expected that no P wave should appear below such frequency. We explain the estimated wavenumbers 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 = 50 m. This is double the thickness of the layer. Therefore the ray theory is not valid and 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 considered wavelengths are too large when compared with the thickness of the layer. It would be interesting to analyze higher frequencies, where wavelengths are shorter and ray theory is applicable. However, the M2.1 dataset is not modeling frequency higher than those considered. 157

0.05 0.07

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Wavenumber [1/m] 0.02 Wavenumber [1/m]

0.01 0.01 0.005

2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Frequency [Hz] Frequency [Hz]

(a) P wave wavenumber κP. Theoretical (b) S wave wavenumber κS. Theoretical P wave wavenumber is depicted with the red S wave wavenumber is depicted with the blue line. line.

1100 1100

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(c) P wave velocity vP. Theoretical P wave (d) S wave velocity vS. Theoretical S wave velocity is depicted with the red line. velocity is depicted with the red line.

0 2π

3π

[rad] 2 η [rad] ψ + π π 4 Inclination Azimuth π 2

+ π 0 2 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 Frequency [Hz] Frequency [Hz] (e) Incidence angle η. (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 single P 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 both quantities. A simple numerical example with a monochromatic wave following the P- SV system shows that the retrieval of all the wavefield parameters of interest is possible. From the shape of the likelihood function (LF), we expect that certain parameters are associated with a large uncertainty. This can be formalized with analysis of Fisher information (FI) and derivation of Cramér-Rao bound (CRB). From the analysis of the M2.1 SESAME synthetic the estimation of parameters of a P wave was not successful due to the limited high-frequency content of 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 sensor arrays for the analysis of seismic surface waves waves that were obtained in Maranò et al. (2013). The arrays described in this supplement are designed for unitary largest wavenumber, i.e., κmax = 1. Guidelines to adapt the physical extent of the array to the actual spatial frequency content of the wavefield are provided in an appendix of the article. The array are obtained using the mixed integer program (MIP) algorithm presented in the article for different number of sensors Ns and different values of κmax/κmin. In particular Ns = 6,..., 20 and κmax/κmin = 2, 4, 6 are considered. We provide sensor coordinates in both Cartesian (xn, yn) and polar coordinates (rn, ψn). The azimuth ψn is measured counter-clockwise from the x-axis.

159 160

0.6 4 2 0.4 [1/m] 1 1

0.2 y κ

0 3 6 0 y [m] −0.2 2 −1

−0.4 Wavenumber −2 −0.6 5

−0.4 −0.2 0 0.2 0.4 0.6 0.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 -0.12564 0.21762 0.25128 120 2 -0.12564 -0.21762 0.25128 240 3 0.25128 0 0.25128 0 4 -0.35897 0.62176 0.71795 120 5 -0.35897 -0.62176 0.71795 240 6 0.71795 0 0.71795 0 (c) Table of coordinates.

Figure D.1: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 6 for κmin = 1/2 and κmax = 1. 161

0.6 2 0.4 1 2

3 [1/m] 0.2 1 y κ

0 7 0 y [m]

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−0.6 −0.4 −0.2 0 0.2 0.4 0.6 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.32249 0.4044 0.51724 51.4 2 -0.1151 0.50427 0.51724 102.9 3 -0.46602 0.22442 0.51724 154.3 4 -0.46602 -0.22442 0.51724 205.7 5 -0.1151 -0.50427 0.51724 257.1 6 0.32249 -0.4044 0.51724 308.6 7 0.51724 0 0.51724 0 (c) Table of coordinates.

Figure D.2: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 7 for κmin = 1/2 and κmax = 1. 162

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5 7 Wavenumber −0.4 −2 6 −0.6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.36472 0.36472 0.51579 45 2 0 0.51579 0.51579 90 3 -0.36472 0.36472 0.51579 135 4 -0.51579 0 0.51579 180 5 -0.36472 -0.36472 0.51579 225 6 0 -0.51579 0.51579 270 7 0.36472 -0.36472 0.51579 315 8 0.51579 0 0.51579 0 (c) Table of coordinates.

Figure D.3: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 8 for κmin = 1/2 and κmax = 1. 163

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[1/m] 1

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−0.6 −0.3 0 0.3 0.6 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.11111 0.19245 0.22222 60 2 -0.22222 0 0.22222 180 3 0.11111 -0.19245 0.22222 300 4 0.33333 0.57735 0.66667 60 5 -0.66667 0 0.66667 180 6 0.33333 -0.57735 0.66667 300 7 -0.045224 0.77646 0.77778 93.3 8 -0.64982 -0.4274 0.77778 213.3 9 0.69505 -0.34907 0.77778 333.3 (c) Table of coordinates.

Figure D.4: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 9 for κmin = 1/2 and κmax = 1. 164

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◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.1553 0.47797 0.50256 72 2 -0.40658 0.2954 0.50256 144 3 -0.40658 -0.2954 0.50256 216 4 0.1553 -0.47797 0.50256 288 5 0.50256 0 0.50256 0 6 0.55179 0.4009 0.68205 36 7 -0.21077 0.64867 0.68205 108 8 -0.68205 0 0.68205 180 9 -0.21077 -0.64867 0.68205 252 10 0.55179 -0.4009 0.68205 324 (c) Table of coordinates.

Figure D.5: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 10 for κmin = 1/2 and κmax = 1. 165

0.9

7 6 2 0.6

3 [1/m] 1

0.3 y 2 κ 0 0 1 y [m]

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−0.6 −0.3 0 0.3 0.6 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.36464 0.21053 0.42105 30 3 -0.21053 0.36464 0.42105 120 4 -0.36464 -0.21053 0.42105 210 5 0.21053 -0.36464 0.42105 300 6 0.19071 0.71173 0.73684 75 7 -0.19071 0.71173 0.73684 105 8 -0.71173 -0.19071 0.73684 195 9 -0.52103 -0.52103 0.73684 225 10 0.52103 -0.52103 0.73684 315 11 0.71173 -0.19071 0.73684 345 (c) Table of coordinates.

Figure D.6: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 11 for κmin = 1/2 and κmax = 1. 166

0.8 10 7 2 0.4 11 4

5 1 [1/m] 1 y

0 3 9 κ

2 0 y [m] −0.4 −1 8 6

−0.8 Wavenumber −2 12

−1.2 −0.8 −0.4 0 0.4 0.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 -0.11111 0.19245 0.22222 120 2 -0.11111 -0.19245 0.22222 240 3 0.22222 0 0.22222 0 4 0.42558 0.3571 0.55556 40 5 -0.52205 0.19001 0.55556 160 6 0.096471 -0.54712 0.55556 280 7 -0.33333 0.57735 0.66667 120 8 -0.33333 -0.57735 0.66667 240 9 0.66667 0 0.66667 0 10 0.76604 0.64279 1 40 11 -0.93969 0.34202 1 160 12 0.17365 -0.98481 1 280 (c) Table of coordinates.

Figure D.7: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 12 for κmin = 1/2 and κmax = 1. 167

0.9 11

0.6 2 9 2 5

0.3 [1/m] 1 y

8 κ

0 6 1 13 0

y [m] 4 −0.3 3 −1 7 Wavenumber −0.6 −2 10 12 −0.9 −0.6 −0.3 0 0.3 0.6 0.9 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 -0.086824 0.4924 0.5 100 3 -0.38302 -0.32139 0.5 220 4 0.46985 -0.17101 0.5 340 5 0.275 0.47631 0.55 60 6 -0.55 0 0.55 180 7 0.275 -0.47631 0.55 300 8 0.67615 0.18117 0.7 15 9 -0.49497 0.49497 0.7 135 10 -0.18117 -0.67615 0.7 255 11 -0.45 0.77942 0.9 120 12 -0.45 -0.77942 0.9 240 13 0.9 0 0.9 0 (c) Table of coordinates.

Figure D.8: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 13 for κmin = 1/2 and κmax = 1. 168

0.8 9 10 2 2 0.4 1

8 [1/m] 1 3 y κ

0 11 7 0 y [m] 4 14 −1 −0.4 6 5 Wavenumber 12 −2 −0.8 13

−0.8 −0.4 0 0.4 0.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.32815 0.41149 0.52632 51.4 2 -0.11712 0.51312 0.52632 102.9 3 -0.47419 0.22836 0.52632 154.3 4 -0.47419 -0.22836 0.52632 205.7 5 -0.11712 -0.51312 0.52632 257.1 6 0.32815 -0.41149 0.52632 308.6 7 0.52632 0 0.52632 0 8 0.75871 0.36538 0.84211 25.7 9 0.18739 0.82099 0.84211 77.1 10 -0.52504 0.65838 0.84211 128.6 11 -0.84211 0 0.84211 180 12 -0.52504 -0.65838 0.84211 231.4 13 0.18739 -0.82099 0.84211 282.9 14 0.75871 -0.36538 0.84211 334.3 (c) Table of coordinates.

Figure D.9: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 14 for κmin = 1/2 and κmax = 1. 169

0.9 13 8 0.6 2

9 2 1 7 0.3 [1/m] 1 y κ

0 3 6 15 0 y [m]

−0.3 −1 10 4 5 12 Wavenumber −0.6 −2 14 11 −0.9 −0.6 −0.3 0 0.3 0.6 0.9 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.225 0.38971 0.45 60 2 -0.225 0.38971 0.45 120 3 -0.45 0 0.45 180 4 -0.225 -0.38971 0.45 240 5 0.225 -0.38971 0.45 300 6 0.45 0 0.45 0 7 0.64952 0.375 0.75 30 8 0 0.75 0.75 90 9 -0.64952 0.375 0.75 150 10 -0.64952 -0.375 0.75 210 11 0 -0.75 0.75 270 12 0.64952 -0.375 0.75 330 13 -0.45 0.77942 0.9 120 14 -0.45 -0.77942 0.9 240 15 0.9 0 0.9 0 (c) Table of coordinates.

Figure D.10: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 15 for κmin = 1/2 and κmax = 1. 170

11 0.8 8 2 14

0.4 3 [1/m] 15 2 1 y

4 κ 0 1 0 9 7 y [m] 5 13 −0.4 6 −1 12 10 Wavenumber −0.8 −2 16

−0.8 −0.4 0 0.4 0.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.33867 0.28418 0.44211 40 3 -0.076771 0.43539 0.44211 100 4 -0.41544 0.15121 0.44211 160 5 -0.33867 -0.28418 0.44211 220 6 0.076771 -0.43539 0.44211 280 7 0.41544 -0.15121 0.44211 340 8 0.27722 0.76165 0.81053 70 9 -0.79821 -0.14075 0.81053 190 10 0.521 -0.6209 0.81053 310 11 -0.15354 0.87078 0.88421 100 12 -0.67734 -0.56836 0.88421 220 13 0.83089 -0.30242 0.88421 340 14 0.73379 0.61572 0.95789 40 15 -0.90013 0.32762 0.95789 160 16 0.16634 -0.94334 0.95789 280 (c) Table of coordinates.

Figure D.11: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 16 for κmin = 1/2 and κmax = 1. 171

0.8 9 12 2

16 1 15 0.4

6 5 [1/m] 1 y κ

0 2 4 14 0 y [m]

7 8 −1 −0.4 10 3 11 Wavenumber 13 −2 −0.8 17

−0.8 −0.4 0 0.4 0.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0.44 0.44 90 2 -0.44 0 0.44 180 3 0 -0.44 0.44 270 4 0.44 0 0.44 0 5 0.33941 0.33941 0.48 45 6 -0.33941 0.33941 0.48 135 7 -0.33941 -0.33941 0.48 225 8 0.33941 -0.33941 0.48 315 9 0 0.8 0.8 90 10 -0.69282 -0.4 0.8 210 11 0.69282 -0.4 0.8 330 12 -0.42 0.72746 0.84 120 13 -0.42 -0.72746 0.84 240 14 0.84 0 0.84 0 15 0.7621 0.44 0.88 30 16 -0.7621 0.44 0.88 150 17 0 -0.88 0.88 270 (c) Table of coordinates.

Figure D.12: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 17 for κmin = 1/2 and κmax = 1. 172

16 0.8 14 2 7 4 10 0.4 [1/m] 1 y

11 1 13 κ

0 2 6 0

y [m] 8 3 18

−0.4 −1 9 5

12 Wavenumber 17 −2 −0.8 15

−0.8 −0.4 0 0.4 0.8 1.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.125 0.21651 0.25 60 2 -0.25 0 0.25 180 3 0.125 -0.21651 0.25 300 4 -0.3 0.51962 0.6 120 5 -0.3 -0.51962 0.6 240 6 0.6 0 0.6 0 7 0.16823 0.62785 0.65 75 8 -0.62785 -0.16823 0.65 195 9 0.45962 -0.45962 0.65 315 10 0.49497 0.49497 0.7 45 11 -0.67615 0.18117 0.7 165 12 0.18117 -0.67615 0.7 285 13 0.92924 0.19752 0.95 12 14 -0.63567 0.70599 0.95 132 15 -0.29357 -0.9035 0.95 252 16 -0.30902 0.95106 1 108 17 -0.66913 -0.74314 1 228 18 0.97815 -0.20791 1 348 (c) Table of coordinates.

Figure D.13: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 18 for κmin = 1/2 and κmax = 1. 173

1.2

17 2 0.8 14 6

7 [1/m] 1 0.4 5 y κ 8 2 1 0 0 15 13 y [m] 3 9 4 19 −1 −0.4 12

10 Wavenumber 11 18 −2 −0.8 16

−0.8 −0.4 0 0.4 0.8 1.2 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.25525 0.14737 0.29474 30 2 -0.14737 0.25525 0.29474 120 3 -0.25525 -0.14737 0.29474 210 4 0.14737 -0.25525 0.29474 300 5 0.50801 0.42627 0.66316 40 6 0.11516 0.65308 0.66316 80 7 -0.33158 0.57431 0.66316 120 8 -0.62316 0.22681 0.66316 160 9 -0.62316 -0.22681 0.66316 200 10 -0.33158 -0.57431 0.66316 240 11 0.11516 -0.65308 0.66316 280 12 0.50801 -0.42627 0.66316 320 13 0.66316 0 0.66316 0 14 0.47895 0.82956 0.95789 60 15 -0.95789 0 0.95789 180 16 0.47895 -0.82956 0.95789 300 17 -0.26699 0.99643 1.0316 105 18 -0.72944 -0.72944 1.0316 225 19 0.99643 -0.26699 1.0316 345 (c) Table of coordinates.

Figure D.14: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 19 for κmin = 1/2 and κmax = 1. 174

16 1 2 12 17 7 0.5 6 [1/m] 2 1 13 y κ 1 11 3 0 20 0 8 y [m] 4 5 10 −1 −0.5 14 18 9 15 Wavenumber −2 −1 19

−1 −0.5 0 0.5 1 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.33657 0.14985 0.36842 24 2 -0.03851 0.3664 0.36842 96 3 -0.36037 0.076599 0.36842 168 4 -0.18421 -0.31906 0.36842 240 5 0.24652 -0.27379 0.36842 312 6 0.44374 0.49282 0.66316 48 7 -0.33158 0.57431 0.66316 120 8 -0.64867 -0.13788 0.66316 192 9 -0.069319 -0.65953 0.66316 264 10 0.60582 -0.26973 0.66316 336 11 0.71369 0.18325 0.73684 14.4 12 0.046267 0.73539 0.73684 86.4 13 -0.6851 0.27125 0.73684 158.4 14 -0.46968 -0.56775 0.73684 230.4 15 0.39482 -0.62214 0.73684 302.4 16 0.34155 1.0512 1.1053 72 17 -0.89418 0.64966 1.1053 144 18 -0.89418 -0.64966 1.1053 216 19 0.34155 -1.0512 1.1053 288 20 1.1053 0 1.1053 0 (c) Table of coordinates.

Figure D.15: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 20 for κmin = 1/2 and κmax = 1. 175

1.2

4 0.8 2

0.4 [1/m] 1 1 y κ

0 3 6 0 y [m] 2 −0.4 −1 Wavenumber −0.8 −2 5

−1.2 −0.8 −0.4 0 0.4 0.8 1.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 -0.15789 0.27348 0.31579 120 2 -0.15789 -0.27348 0.31579 240 3 0.31579 0 0.31579 0 4 -0.57895 1.0028 1.1579 120 5 -0.57895 -1.0028 1.1579 240 6 1.1579 0 1.1579 0 (c) Table of coordinates.

Figure D.16: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 6 for κmin = 1/4 and κmax = 1. 176

1.8

5 2 1.2

[1/m] 1 y

0.6 κ 2 0 y [m] 0 1 −1 3 4 Wavenumber −0.6 −2 6 7

−1.2 −0.6 0 0.6 1.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0 0.5 0.5 90 3 -0.43301 -0.25 0.5 210 4 0.43301 -0.25 0.5 330 5 0 1.5 1.5 90 6 -1.299 -0.75 1.5 210 7 1.299 -0.75 1.5 330 (c) Table of coordinates.

Figure D.17: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 7 for κmin = 1/4 and κmax = 1. 177

5 1 2 4

0.5 [1/m] 1 y

1 κ 0 6 2 0

y [m] 3

−0.5 −1

8 Wavenumber −1 −2 7

−1.5 −1 −0.5 0 0.5 1 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.16904 0.18774 0.25263 48 2 -0.24711 0.052525 0.25263 168 3 0.078067 -0.24027 0.25263 288 4 1.0219 0.74247 1.2632 36 5 -0.39034 1.2013 1.2632 108 6 -1.2632 0 1.2632 180 7 -0.39034 -1.2013 1.2632 252 8 1.0219 -0.74247 1.2632 324 (c) Table of coordinates.

Figure D.18: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 8 for κmin = 1/4 and κmax = 1. 178

4 1 2 7 1

0.5 [1/m] 8 1 y κ

0 0

y [m] 2 −0.5 5 3 −1 6 Wavenumber −1 −2 9

−1.5 −1 −0.5 0 0.5 1 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.13161 0.74638 0.75789 80 2 -0.71219 -0.25922 0.75789 200 3 0.58058 -0.48717 0.75789 320 4 0.088103 1.1757 1.1789 85.7 5 -1.0622 -0.51153 1.1789 205.7 6 0.97409 -0.66412 1.1789 325.7 7 0.96764 0.81194 1.2632 40 8 -1.187 0.43203 1.2632 160 9 0.21935 -1.244 1.2632 280 (c) Table of coordinates.

Figure D.19: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 9 for κmin = 1/4 and κmax = 1. 179

1.2 9 5 2

0.6 [1/m] 1

2 y 8 κ 0 1 4 0

y [m] 3 7 −0.6 6 −1 Wavenumber −1.2 −2 10

−1.2 −0.6 0 0.6 1.2 1.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 -0.1931 0.33446 0.38621 120 3 -0.1931 -0.33446 0.38621 240 4 0.38621 0 0.38621 0 5 -0.12688 1.2071 1.2138 96 6 -0.98198 -0.71345 1.2138 216 7 1.1089 -0.49369 1.2138 336 8 1.5844 0.22268 1.6 8 9 -0.98506 1.2608 1.6 128 10 -0.59937 -1.4835 1.6 248 (c) Table of coordinates.

Figure D.20: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 10 for κmin = 1/4 and κmax = 1. 180

8 1 2 7 2 3 0.5 [1/m] 1 y κ 0 1 6 0

y [m] 9

−0.5 11 4 −1 Wavenumber −1 5 −2 10

−1 −0.5 0 0.5 1 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.31227 0.96107 1.0105 72 3 -0.81753 0.59397 1.0105 144 4 -0.81753 -0.59397 1.0105 216 5 0.31227 -0.96107 1.0105 288 6 1.0105 0 1.0105 0 7 0.80296 0.89177 1.2 48 8 -0.6 1.0392 1.2 120 9 -1.1738 -0.24949 1.2 192 10 -0.12543 -1.1934 1.2 264 11 1.0963 -0.48808 1.2 336 (c) Table of coordinates.

Figure D.21: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 11 for κmin = 1/4 and κmax = 1. 181

1.2 10 4 2 1 7

0.6 8 [1/m] 1 y

11 κ 0 0

y [m] 2 6 −0.6 3 5 −1 Wavenumber −1.2 −2 9 12

−1.8 −1.2 −0.6 0 0.6 1.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.15435 0.87538 0.88889 80 2 -0.83528 -0.30402 0.88889 200 3 0.68093 -0.57137 0.88889 320 4 -0.19294 1.0942 1.1111 100 5 -0.85116 -0.71421 1.1111 220 6 1.0441 -0.38002 1.1111 340 7 1.0214 0.85705 1.3333 40 8 -1.2529 0.45603 1.3333 160 9 0.23153 -1.3131 1.3333 280 10 0.92891 1.2477 1.5556 53.3 11 -1.545 0.18059 1.5556 173.3 12 0.61612 -1.4283 1.5556 293.3 (c) Table of coordinates.

Figure D.22: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 12 for κmin = 1/4 and κmax = 1. 182

1.2 8 11 2 2 5 0.6 12 6 [1/m] 1 y κ 0 1 4 10 0 y [m] −0.6 3 −1

9 7 −1.2 Wavenumber −2 13 −1.8 −1.2 −0.6 0 0.6 1.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 -0.44444 0.7698 0.88889 120 3 -0.44444 -0.7698 0.88889 240 4 0.88889 0 0.88889 0 5 0.85116 0.71421 1.1111 40 6 -1.0441 0.38002 1.1111 160 7 0.19294 -1.0942 1.1111 280 8 -0.66667 1.1547 1.3333 120 9 -0.66667 -1.1547 1.3333 240 10 1.3333 0 1.3333 0 11 1.1916 0.99989 1.5556 40 12 -1.4617 0.53203 1.5556 160 13 0.27012 -1.5319 1.5556 280 (c) Table of coordinates.

Figure D.23: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 13 for κmin = 1/4 and κmax = 1. 183

9 1.4 8 2

0.7 10 [1/m] 2 1

1 y 3 κ 0 7 14 0

y [m] 4 6 5 −0.7 11 −1 Wavenumber 13 −2 −1.4 12

−1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.38699 0.48527 0.62069 51.4 2 -0.13812 0.60513 0.62069 102.9 3 -0.55922 0.26931 0.62069 154.3 4 -0.55922 -0.26931 0.62069 205.7 5 -0.13812 -0.60513 0.62069 257.1 6 0.38699 -0.48527 0.62069 308.6 7 0.62069 0 0.62069 0 8 1.032 1.2941 1.6552 51.4 9 -0.36831 1.6137 1.6552 102.9 10 -1.4913 0.71815 1.6552 154.3 11 -1.4913 -0.71815 1.6552 205.7 12 -0.36831 -1.6137 1.6552 257.1 13 1.032 -1.2941 1.6552 308.6 14 1.6552 0 1.6552 0 (c) Table of coordinates.

Figure D.24: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 14 for κmin = 1/4 and κmax = 1. 184

1.4 10 14 4 5 2 0.7 13 2 1 [1/m] 1 y

0 9 κ 11 8 0

y−0.7 [m] 6 3 12 7 −1

−1.4 Wavenumber −2 15 −2.1 −1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.78705 0.35042 0.86154 24 2 -0.697 0.5064 0.86154 144 3 -0.090055 -0.85682 0.86154 264 4 -0.11257 1.071 1.0769 96 5 -0.53846 0.93264 1.0769 120 6 -0.87125 -0.633 1.0769 216 7 -0.53846 -0.93264 1.0769 240 8 0.98382 -0.43802 1.0769 336 9 1.0769 0 1.0769 0 10 0.37716 1.1608 1.2205 72 11 -1.1938 -0.25376 1.2205 192 12 0.81668 -0.90702 1.2205 312 13 1.7053 0.75924 1.8667 24 14 -1.5102 1.0972 1.8667 144 15 -0.19512 -1.8564 1.8667 264 (c) Table of coordinates.

Figure D.25: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 15 for κmin = 1/4 and κmax = 1. 185

1.6 13 12 8 2 0.8 5

14 11 [1/m] 1

2 y κ 0 3 1 7 10 4 0 y [m] −0.8 6 −1 9 Wavenumber −1.6 −2 15 16

−1.6 −0.8 0 0.8 1.6 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.16923 0.29312 0.33846 60 3 -0.33846 0 0.33846 180 4 0.16923 -0.29312 0.33846 300 5 -0.50769 0.87935 1.0154 120 6 -0.50769 -0.87935 1.0154 240 7 1.0154 0 1.0154 0 8 -0.67692 1.1725 1.3538 120 9 -0.67692 -1.1725 1.3538 240 10 1.3538 0 1.3538 0 11 1.7981 0.4818 1.8615 15 12 1.3163 1.3163 1.8615 45 13 -1.3163 1.3163 1.8615 135 14 -1.7981 0.4818 1.8615 165 15 -0.4818 -1.7981 1.8615 255 16 0.4818 -1.7981 1.8615 285 (c) Table of coordinates.

Figure D.26: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 16 for κmin = 1/4 and κmax = 1. 186

11 1.4 2 12 6 13 10 0.7 [1/m] 1 9 y 3 2 κ 0 1 8 0

y [m] 4 5 17

−0.7 −1

14 Wavenumber −2 −1.4 15 16

−1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.24244 0.24244 0.34286 45 3 -0.24244 0.24244 0.34286 135 4 -0.24244 -0.24244 0.34286 225 5 0.24244 -0.24244 0.34286 315 6 -0.6 1.0392 1.2 120 7 -0.6 -1.0392 1.2 240 8 1.2 0 1.2 0 9 1.4903 0.39932 1.5429 15 10 1.3362 0.77143 1.5429 30 11 -0.39932 1.4903 1.5429 105 12 -1.091 1.091 1.5429 135 13 -1.3362 0.77143 1.5429 150 14 -1.091 -1.091 1.5429 225 15 -0.39932 -1.4903 1.5429 255 16 0 -1.5429 1.5429 270 17 1.4903 -0.39932 1.5429 345 (c) Table of coordinates.

Figure D.27: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 17 for κmin = 1/4 and κmax = 1. 187

14 1.4 8 7 2 15 4 0.7 [1/m] 13 1 1 y κ

0 2 0

y [m] 6 9 3 12 5 18 −0.7 −1 10 11 16 Wavenumber −1.4 −2 17

−1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.25128 0.43523 0.50256 60 2 -0.50256 0 0.50256 180 3 0.25128 -0.43523 0.50256 300 4 -0.13714 0.77775 0.78974 100 5 -0.60498 -0.50764 0.78974 220 6 0.74212 -0.27011 0.78974 340 7 0.21194 1.202 1.2205 80 8 -0.21194 1.202 1.2205 100 9 -1.1469 -0.41744 1.2205 200 10 -0.93497 -0.78453 1.2205 220 11 0.93497 -0.78453 1.2205 320 12 1.1469 -0.41744 1.2205 340 13 1.5517 0.56477 1.6513 20 14 -0.28674 1.6262 1.6513 100 15 -1.265 1.0614 1.6513 140 16 -1.265 -1.0614 1.6513 220 17 -0.28674 -1.6262 1.6513 260 18 1.5517 -0.56477 1.6513 340 (c) Table of coordinates.

Figure D.28: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 18 for κmin = 1/4 and κmax = 1. 188

1.4 10 9 8 17 2 5 0.7 2

18 [1/m] 1 y κ 0 11 3 1 7 0

y [m] 12 16

−0.7 4 −1 13 6 15

14 Wavenumber −1.4 −2 19

−1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.44211 0.76575 0.88421 60 3 -0.88421 0 0.88421 180 4 0.44211 -0.76575 0.88421 300 5 -0.51579 0.89337 1.0316 120 6 -0.51579 -0.89337 1.0316 240 7 1.0316 0 1.0316 0 8 0.66316 1.1486 1.3263 60 9 0.23031 1.3062 1.3263 80 10 -0.23031 1.3062 1.3263 100 11 -1.3263 0 1.3263 180 12 -1.2463 -0.45363 1.3263 200 13 -1.016 -0.85254 1.3263 220 14 0.66316 -1.1486 1.3263 300 15 1.016 -0.85254 1.3263 320 16 1.2463 -0.45363 1.3263 340 17 1.1463 1.1463 1.6211 45 18 -1.5658 0.41956 1.6211 165 19 0.41956 -1.5658 1.6211 285 (c) Table of coordinates.

Figure D.29: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 19 for κmin = 1/4 and κmax = 1. 189

13 1.4 12 14 6 2

0.7 7 2 15 [1/m] 1 11 y 1 κ 3 0 10 0 y [m] 20 16 5 −1 −0.7 8 4

17 9 Wavenumber 19 −2 −1.4 18

−1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.67314 0.2997 0.73684 24 2 -0.077021 0.73281 0.73684 96 3 -0.72074 0.1532 0.73684 168 4 -0.36842 -0.63812 0.73684 240 5 0.49304 -0.54758 0.73684 312 6 0.36431 1.1212 1.1789 72 7 -0.95379 0.69297 1.1789 144 8 -0.95379 -0.69297 1.1789 216 9 0.36431 -1.1212 1.1789 288 10 1.1789 0 1.1789 0 11 1.4274 0.36649 1.4737 14.4 12 0.78964 1.2443 1.4737 57.6 13 0.092533 1.4708 1.4737 86.4 14 -0.93936 1.1355 1.4737 129.6 15 -1.3702 0.5425 1.4737 158.4 16 -1.3702 -0.5425 1.4737 201.6 17 -0.93936 -1.1355 1.4737 230.4 18 0.092533 -1.4708 1.4737 273.6 19 0.78964 -1.2443 1.4737 302.4 20 1.4274 -0.36649 1.4737 345.6 (c) Table of coordinates.

Figure D.30: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 20 for κmin = 1/4 and κmax = 1. 190

2 1 2 3

0.5 [1/m] 1 y κ

0 1 6 0 y [m]

−0.5 −1 4 Wavenumber −2 −1 5

−1 −0.5 0 0.5 1 1.5 −2 −1 0 1 2 x [m] Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.38058 1.1713 1.2316 72 3 -0.99637 0.7239 1.2316 144 4 -0.99637 -0.7239 1.2316 216 5 0.38058 -1.1713 1.2316 288 6 1.2316 0 1.2316 0 (c) Table of coordinates.

Figure D.31: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 6 for κmin = 1/6 and κmax = 1. 191

1.8 5 2 1.2 2 [1/m] 1 y

0.6 κ 0 y [m] 0 1 −1

3 4 −0.6 Wavenumber −2 6 7

−1.2 −1.2 −0.6 0 0.6 1.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0 1.0345 1.0345 90 3 -0.89589 -0.51724 1.0345 210 4 0.89589 -0.51724 1.0345 330 5 0 1.5517 1.5517 90 6 -1.3438 -0.77586 1.5517 210 7 1.3438 -0.77586 1.5517 330 (c) Table of coordinates.

Figure D.32: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 7 for κmin = 1/6 and κmax = 1. 192

3 1.4 2 2

0.7 4 [1/m] 1 y κ

0 1 8 0 y [m]

−0.7 5 −1 Wavenumber 7 −2 −1.4 6

−1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.96477 1.2098 1.5474 51.4 3 -0.34432 1.5086 1.5474 102.9 4 -1.3941 0.67138 1.5474 154.3 5 -1.3941 -0.67138 1.5474 205.7 6 -0.34432 -1.5086 1.5474 257.1 7 0.96477 -1.2098 1.5474 308.6 8 1.5474 0 1.5474 0 (c) Table of coordinates.

Figure D.33: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 8 for κmin = 1/6 and κmax = 1. 193

1.4 7 1 4 2 0.7 8 5 [1/m] 1 y

0 3 κ 0 y [m] −0.7 2 −1

−1.4 Wavenumber −2 9 −2.1 −1.4 −0.7 0 0.7 1.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 -0.53846 0.93264 1.0769 120 2 -0.53846 -0.93264 1.0769 240 3 1.0769 0 1.0769 0 4 1.0725 0.8999 1.4 40 5 -1.3156 0.47883 1.4 160 6 0.24311 -1.3787 1.4 280 7 1.4025 1.1768 1.8308 40 8 -1.7204 0.62616 1.8308 160 9 0.31791 -1.803 1.8308 280 (c) Table of coordinates.

Figure D.34: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 9 for κmin = 1/6 and κmax = 1. 194

2.7 8 2 1.8

5 [1/m] 1 y

0.9 κ

2 0 y [m] 0 1 3 4 −1 7 6 −0.9 Wavenumber 9 −2 10

−1.8 −1.8 −0.9 0 0.9 1.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.15344 0.47225 0.49655 72 3 -0.4857 -0.10324 0.49655 192 4 0.33226 -0.36901 0.49655 312 5 -0.14706 1.3992 1.4069 96 6 -1.1382 -0.82695 1.4069 216 7 1.2853 -0.57224 1.4069 336 8 0.083759 2.3985 2.4 88 9 -2.1191 -1.1267 2.4 208 10 2.0353 -1.2718 2.4 328 (c) Table of coordinates.

Figure D.35: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 10 for κmin = 1/6 and κmax = 1. 195

9 1.8 6 2 [1/m] 0.9 1 1 y κ 2 0 0 5 y [m] 7 10 3 −1 4 −0.9 Wavenumber 8 −2 11 −1.8

−1.8 −0.9 0 0.9 1.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.24396 0.75083 0.78947 72 2 -0.6387 0.46404 0.78947 144 3 -0.6387 -0.46404 0.78947 216 4 0.24396 -0.75083 0.78947 288 5 0.78947 0 0.78947 0 6 0.53671 1.6518 1.7368 72 7 -1.6989 -0.36111 1.7368 192 8 1.1622 -1.2907 1.7368 312 9 0.68309 2.1023 2.2105 72 10 -2.1622 -0.45959 2.2105 192 11 1.4791 -1.6427 2.2105 312 (c) Table of coordinates.

Figure D.36: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 11 for κmin = 1/6 and κmax = 1. 196

10 2.2 2

1.1 8 7 [1/m] 4 1 y κ 2 1 0 11 5 0

y [m] 3 6 −1 −1.1 Wavenumber −2 9 −2.2 12

−2.2 −1.1 0 1.1 2.2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.46632 0.26923 0.53846 30 2 -0.46632 0.26923 0.53846 150 3 0 -0.53846 0.53846 270 4 0.48462 0.83938 0.96923 60 5 -0.96923 0 0.96923 180 6 0.48462 -0.83938 0.96923 300 7 1.772 1.0231 2.0462 30 8 -1.772 1.0231 2.0462 150 9 0 -2.0462 2.0462 270 10 1.3462 2.3316 2.6923 60 11 -2.6923 0 2.6923 180 12 1.3462 -2.3316 2.6923 300 (c) Table of coordinates.

Figure D.37: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 12 for κmin = 1/6 and κmax = 1. 197

1.8

12 9 3 11 2 0.9 2 8 4 [1/m] 1 0 1 y κ

5 0

y−0.9 [m] 7 6 −1

−1.8 10 Wavenumber −2 13 −2.7 −1.8 −0.9 0 0.9 1.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 1.0214 0.85705 1.3333 40 3 0.23153 1.3131 1.3333 80 4 -1.2529 0.45603 1.3333 160 5 -1.2529 -0.45603 1.3333 200 6 0.23153 -1.3131 1.3333 280 7 1.0214 -0.85705 1.3333 320 8 1.8794 0.68404 2 20 9 -1.5321 1.2856 2 140 10 -0.3473 -1.9696 2 260 11 2.0851 1.0472 2.3333 26.7 12 -1.9495 1.2822 2.3333 146.7 13 -0.13567 -2.3294 2.3333 266.7 (c) Table of coordinates.

Figure D.38: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 13 for κmin = 1/6 and κmax = 1. 198

9 1.8 3 2 8 2

0.9 10 [1/m] 1 y

4 1 κ 0 14 0 y [m]

−0.9 11 −1 5 7 Wavenumber 13 −2 −1.8 6 12

−1.8 −0.9 0 0.9 1.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 1.7549 0.40054 1.8 12.9 2 0.78099 1.6217 1.8 64.3 3 -0.78099 1.6217 1.8 115.7 4 -1.7549 0.40054 1.8 167.1 5 -1.4073 -1.1223 1.8 218.6 6 0 -1.8 1.8 270 7 1.4073 -1.1223 1.8 321.4 8 1.247 1.5637 2 51.4 9 -0.44504 1.9499 2 102.9 10 -1.8019 0.86777 2 154.3 11 -1.8019 -0.86777 2 205.7 12 -0.44504 -1.9499 2 257.1 13 1.247 -1.5637 2 308.6 14 2 0 2 0 (c) Table of coordinates.

Figure D.39: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 14 for κmin = 1/6 and κmax = 1. 199

2 8

14 7 2 1 13 9 4 1

[1/m] 1 y

0 6 12 κ 3 0 2 y [m]

−1 5 −1

10 Wavenumber −2 11 −2 15

−2 −1 0 1 2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 -0.091225 0.86795 0.87273 96 2 -0.70605 -0.51298 0.87273 216 3 0.79728 -0.35497 0.87273 336 4 -0.54545 0.94475 1.0909 120 5 -0.54545 -0.94475 1.0909 240 6 1.0909 0 1.0909 0 7 1.7651 1.2824 2.1818 36 8 -1.0909 1.8895 2.1818 120 9 -1.9932 0.88743 2.1818 156 10 -1.0909 -1.8895 2.1818 240 11 0.22806 -2.1699 2.1818 276 12 2.1818 0 2.1818 0 13 2.1925 0.97617 2.4 24 14 -1.9416 1.4107 2.4 144 15 -0.25087 -2.3869 2.4 264 (c) Table of coordinates.

Figure D.40: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 15 for κmin = 1/6 and κmax = 1. 200

1.8 7 6 5 14 2

0.9 2 [1/m] 1 3 y κ

0 15 1 0 y [m] 8 13 −1 −0.9 9 12 10 11 Wavenumber 4 −2 −1.8 16

−1.8 −0.9 0 0.9 1.8 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 1.1667 0.97901 1.5231 40 3 -1.4312 0.52092 1.5231 160 4 0.26448 -1.4999 1.5231 280 5 0.4599 1.7164 1.7769 75 6 0 1.7769 1.7769 90 7 -0.4599 1.7164 1.7769 105 8 -1.7164 -0.4599 1.7769 195 9 -1.5389 -0.88846 1.7769 210 10 -1.2565 -1.2565 1.7769 225 11 1.2565 -1.2565 1.7769 315 12 1.5389 -0.88846 1.7769 330 13 1.7164 -0.4599 1.7769 345 14 1.0154 1.7587 2.0308 60 15 -2.0308 0 2.0308 180 16 1.0154 -1.7587 2.0308 300 (c) Table of coordinates.

Figure D.41: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 16 for κmin = 1/6 and κmax = 1. 201

2.4 10 11 2 12 1.2 5 9 [1/m] 1 y

4 κ 6 1 0 3 17 2 0

y [m] 13 8 7 −1.2 −1

16 Wavenumber 14 −2 −2.4 15

−2.4 −1.2 0 1.2 2.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 -0.125 0.21651 0.25 120 2 -0.125 -0.21651 0.25 240 3 0.25 0 0.25 0 4 1.1419 0.50842 1.25 24 5 -0.13066 1.2432 1.25 96 6 -1.2227 0.25989 1.25 168 7 -0.625 -1.0825 1.25 240 8 0.83641 -0.92893 1.25 312 9 2.2839 1.0168 2.5 24 10 0.77254 2.3776 2.5 72 11 -1.25 2.1651 2.5 120 12 -2.0225 1.4695 2.5 144 13 -2.4454 -0.51978 2.5 192 14 -1.25 -2.1651 2.5 240 15 -0.26132 -2.4863 2.5 264 16 1.6728 -1.8579 2.5 312 17 2.5 0 2.5 0 (c) Table of coordinates.

Figure D.42: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 17 for κmin = 1/6 and κmax = 1. 202

16 2 13 8 2 4

1 9 7 [1/m] 1 y

1 κ

0 2 6 15 18 0

y [m] 3

−1 10 12 −1

5 Wavenumber 14 11 −2 −2 17

−2 −1 0 1 2 3 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0.225 0.38971 0.45 60 2 -0.45 0 0.45 180 3 0.225 -0.38971 0.45 300 4 -0.825 1.4289 1.65 120 5 -0.825 -1.4289 1.65 240 6 1.65 0 1.65 0 7 1.5588 0.9 1.8 30 8 0 1.8 1.8 90 9 -1.5588 0.9 1.8 150 10 -1.5588 -0.9 1.8 210 11 0 -1.8 1.8 270 12 1.5588 -0.9 1.8 330 13 -1.05 1.8187 2.1 120 14 -1.05 -1.8187 2.1 240 15 2.1 0 2.1 0 16 -1.275 2.2084 2.55 120 17 -1.275 -2.2084 2.55 240 18 2.55 0 2.55 0 (c) Table of coordinates.

Figure D.43: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 18 for κmin = 1/6 and κmax = 1. 203

17 2.4 2 11 14

1.2 [1/m] 1

7 6 y κ 8 5 15 2 3 0 0 12 1 y [m] 4 19 −1 −1.2 9 10 Wavenumber 18 13 −2 16 −2.4

−2.4 −1.2 0 1.2 2.4 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 0 0 0 0 2 0.45963 0.38567 0.6 40 3 -0.56382 0.20521 0.6 160 4 0.10419 -0.59088 0.6 280 5 1.4095 0.51303 1.5 20 6 1.1491 0.96418 1.5 40 7 -1.1491 0.96418 1.5 140 8 -1.4095 0.51303 1.5 160 9 -0.26047 -1.4772 1.5 260 10 0.26047 -1.4772 1.5 280 11 1.05 1.8187 2.1 60 12 -2.1 0 2.1 180 13 1.05 -1.8187 2.1 300 14 1.4964 1.8764 2.4 51.4 15 -2.3732 0.3577 2.4 171.4 16 0.87682 -2.2341 2.4 291.4 17 -0.60081 2.6323 2.7 102.9 18 -1.9792 -1.8365 2.7 222.9 19 2.58 -0.79584 2.7 342.9 (c) Table of coordinates.

Figure D.44: Array layout, magnitude of the Fourier transform of the sensor positions, and sensor coordinates for Ns = 19 for κmin = 1/6 and κmax = 1. 204

17 2 2 12 7 11 2

1 [1/m] 6 1 18 16 y 1 κ 3 0 8 0 y [m]

13 15 10 −1 −1 5 4

9 Wavenumber 14 −2 −2 19 20

−2 −1 0 1 2 x [m] −2 −1 0 1 2 Wavenumber κx [1/m] 2 (a) h(x, y) (b) |H(κx, κy)|

◦ n xn [m] yn [m] rn [m] ψn [ ] 1 1.1746 0.52295 1.2857 24 2 -0.13439 1.2787 1.2857 96 3 -1.2576 0.26732 1.2857 168 4 -0.64286 -1.1135 1.2857 240 5 0.86031 -0.95547 1.2857 312 6 1.2135 0.88168 1.5 36 7 -0.46353 1.4266 1.5 108 8 -1.5 0 1.5 180 9 -0.46353 -1.4266 1.5 252 10 1.2135 -0.88168 1.5 324 11 1.0076 1.3869 1.7143 54 12 -1.0076 1.3869 1.7143 126 13 -1.6304 -0.52974 1.7143 198 14 0 -1.7143 1.7143 270 15 1.6304 -0.52974 1.7143 342 16 2.2418 0.7284 2.3571 18 17 0 2.3571 2.3571 90 18 -2.2418 0.7284 2.3571 162 19 -1.3855 -1.907 2.3571 234 20 1.3855 -1.907 2.3571 306 (c) Table of coordinates.

Figure D.45: Array layout, magnitude of the Fourier transform of the sensor positions, 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 parameters of 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 the time instants tk on the ℓ-th channel is modeled as a random variable

(ℓ) (ℓ) (ℓ) Yk = uk (θ) + Zk , (E.1)

(ℓ) where uk (θ) is a deterministic function of the waveﬁeld parameters θ and (ℓ) (ℓ) 2 Zk represents additive Gaussian noise, Zk 0, σℓ . Let K the number of sampling times t and L the number of channels.∼ N k Assuming independent noise, the probability density function (PDF) of the (ℓ) measurements Y = Yk k=1,...,K is given by the product of Gaussian densities { } ℓ=1,...,L

L K (ℓ) (ℓ) 2 1 (yk uk (θ)) pY (y θ) = exp − , (E.2) 2 2σ2 | 2πσℓ − ℓ ! ℓY=1 kY=1 and its logarithm is p

L L K LK K 2 1 (ℓ) (ℓ) 2 ln pY (y θ) = ln(2π) ln σℓ 2 (Yk uk ) . (E.3) | − 2 − 2 − 2σℓ − Xℓ=1 Xℓ=1 Xk=1 Fisher Information with Known Noise Variance The FIM is deﬁned Fisher (1922); Rao (1945); Cramér (1946) as

2 ∂ ln pY (y θ) I(θ) = E | . (E.4) − ∂θ2

205 206

We assume the variances be known and not dependent on θ, then the element in position i j of the FIM is

2 ∂ ln pY (y θ) [I(θ)] = E | (E.5) i,j − ∂θ ∂θ i j 2 L K E ∂ (ℓ) (ℓ) 2 = ln Yk uk , σℓ (E.6) (−∂θi∂θj N | ) Xℓ=1 Xk=1 2 L K E ∂ 1 (ℓ) (ℓ) 2 = + 2 (Yk uk ) (E.7) ( ∂θi∂θj 2σℓ − ) Xℓ=1 Xk=1 L K (ℓ) E ∂ 1 (ℓ) (ℓ) ∂uk = 2 (Yk uk ) (E.8) (−∂θj σℓ − ∂θi ) Xℓ=1 Xk=1 L K (ℓ) (ℓ) 2 (ℓ) E 1 ∂uk ∂uk (ℓ) (ℓ) ∂ uk = 2 + (Yk uk ) (E.9) (− σℓ − ∂θi ∂θj − ∂θi∂θj !) Xℓ=1 Xk=1 L K (ℓ) (ℓ) 1 ∂uk ∂uk = 2 . (E.10) σℓ ∂θi ∂θj Xℓ=1 Xk=1 The FIM is a positive semideﬁnite (PSD) matrix of size as the number of elements of the parameter θ,

Iθ1,θ1 Iθ2,θ1 ··· I θ ,θ θ ,θ (θ) =  I 1 2 I 2 2 ···  . (E.11) . . .  . . ..      Fisher Information with Unknown Noise Variance The setting of unknown noise variance is now considered. It will be shown that this new case is accounted for with simple modifications from the known variance setting. We distinguish two cases. In the first case, all channels share the same noise variance σ2. In the second case, each channel has different noise variance, 2 2 2 σ1, σ2 , . . . , σL.

Equal Noise Variance on Each Channel In the ﬁrst case, we consider the extended parameter vector η = (θT, σ2)T. The partial derivatives are

L K (ℓ) ∂ ln pY (y η) 1 (ℓ) (ℓ) ∂uk | = 2 (Yk uk ) (E.12) ∂θi −σ − ∂θi Xℓ=1 kX=1 L K ∂ ln pY (y η) KL 1 | = + (Y (ℓ) u(ℓ))2 . (E.13) ∂σ2 − 2σ2 2σ4 k − k Xℓ=1 Xk=1 207

It follows that the FIM elements are

L K (ℓ) (ℓ) 1 ∂uk ∂uk θi,θj = 2 (E.14) I σ ∂θi ∂θj Xℓ=1 Xk=1 L K (ℓ) 1 (ℓ) ∂uk 2 E θi,σ = 4 Zk I (2σ ∂θi Xℓ=1 kX=1 L K L K (ℓ2) 1 ∂u (Z(ℓ1))2Z(ℓ2) k2 (E.15) 6 k1 k2 −2σ ∂θi ) ℓX1=1 kX1=1 ℓX2=1 kX2=1 = 0 (E.16)

2 2 L K K L KL (ℓ) 2 2 2 E σ ,σ = 4 6 (Zk ) I ( 4σ − 4σ Xℓ=1 kX=1 L K L K 1 + (Z(ℓ1))2(Z(ℓ2))2 (E.17) 8 k1 k2 4σ ) ℓX1=1 kX1=1 ℓX2=1 kX2=1 K2L2 K2L2 3LK LK(LK 1) = + + − (E.18) 4σ4 − 2σ4 4σ4 4σ4 KL = , (E.19) 2σ4 where we used the fact that the third moment is E (Z(ℓ))3 = 0 and the fourth { k } E (ℓ) 4 4 moment is (Zk ) = 3σ . The FIM{ is }

I(θ) 0 I(η) = . (E.20) 0 2 2 Iσ ,σ ! The FIM is block diagonal. This implies that the waveﬁeld parameters θ and the variance are decoupled, i.e., the uncertainty about θi does not aﬀect the estimation 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 each channel. We consider the extended parameter vector η = (θT, σT)T with 2 2 2 T σ = (σ1 , σ2 , . . . , σL) . The partial derivatives are

L K (ℓ) ∂ ln pY (y η) 1 (ℓ) (ℓ) ∂uk | = 2 (Yk uk ) (E.21) ∂θi − σℓ − ∂θi Xℓ=1 Xk=1 K ∂ ln pY (y η) K 1 (ℓ) (ℓ) 2 2 | = 2 + 4 (Yk uk ) . (E.22) ∂σℓ −2σℓ 2σℓ − kX=1 208

It follows that each element of the FIM is L K (ℓ) (ℓ) 1 ∂uk ∂uk θi,θj = 2 (E.23) I σℓ ∂θi ∂θj Xℓ=1 Xk=1 L K (ℓ) 1 (ℓ) ∂uk 2 = E Z θi,σℓ 4 k I (2σ ∂θi Xℓ=1 Xk=1 L K L K (ℓ2) 1 ∂u (Z(ℓ1))2Z(ℓ2) k2 (E.24) 6 k1 k2 −2σ ∂θi ) ℓX1=1 kX1=1 ℓX2=1 kX2=1 = 0 (E.25)

2 K K K K K (ℓ) 2 1 (ℓ) 2 (ℓ) 2 2 2 = E (Z ) + (Z ) (Z ) (E.26) σℓ ,σℓ 4 6 k 8 k1 k2 I (4σℓ − 4σℓ 4σℓ ) kX=1 kX1=1 kX2=1 K2 K2 3K K(K 1) = 4 4 + 4 + −4 (E.27) 4σℓ − 4σℓ 4σℓ 4σℓ K = (E.28) 2σ4 2 K (ℓ) 2 K (p) 2 K (p) 2 E K k=1(Zk ) k=1(Zk ) K k=1(Zk ) σ2,σ2 = + I ℓ p 4σ2σ2 4σ4, σ4 − 4σ2σ4 ( ℓ p P ℓ Pp P ℓ p K K (Z(ℓ))2 k=1 k (E.29) − 4σ2σ4 P p ℓ ) K2 K2 K2 K2 = 2 2 + 2 2 2 2 2 2 (E.30) 4σℓ σp 4σℓ , σp − 4σℓ σp − 4σℓ σp = 0 . (E.31) The FIM is

I(θ) 0 0 0

0 σ2,σ2 0 0  1 1  I(η) = I . (E.32) ..  0 0 . 0     0 0 0 σ2 ,σ2   L L   I  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 block diagonal. Therefore we proceed with our further derivations assuming known variance. It is immediate to extend the results by augmenting appropriately the FIMs.

E.2 Derivation of Fisher Information Matrices

In this section we derive the FIMs for diﬀerent setups. We consider the statistical models involving a single Love wave or a single Rayleigh wave using 209 translational measurements, rotational measurements, or both types of measurements.

Approximations Throughout the document we use the following approximations

K K cos2(ωk + ϕ) (E.33) ≈ 2 kX=1 K K sin2(ωk + ϕ) (E.34) ≈ 2 Xk=1 K sin(ωk + ϕ) cos(ωk + ϕ) 0 , (E.35) ≈ Xk=1 2πm Z which are valid for ω being not near 0 or 1/2 and are exact when ω = K m (see (Kay, 1993, example 3.14) or Stoica et al. (1989)). ∈

Useful Quantities We deﬁne Φ = x κ cos ψ y κ sin ψ + ϕ, then we have n − n − n

∂Φ n = x cos ψ y sin ψ (E.36) ∂κ − n − n ∂Φ n = +x κ sin ψ y κ cos ψ (E.37) ∂ψ n − n ∂Φ n = 1 . (E.38) ∂ϕ

Parameter Space Transformation Let θ and η be two diﬀerent parameterizations of a statistical model. The two FIMs are related as

I(η) = JTI(θ(η))J , (E.39) J J where is the Jacobian matrix with elements [ ]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 wavenumber κ′ = κ/2π measured in 1/m should use the parameters space transformation as in (E.39). In practice, let η = (α, ϕ, κ/2π, ψ)T

1 0 0 0 0 1 0 0 J =   . (E.40) 0 0 2π 0    0 0 0 1      210

Moreover it is also useful, to the aim of computing the CRB of a subset of the parameters, to permute the order of the parameters. For example, to move the wavenumber in the ﬁrst position one can use the transformation of the FIM

0 0 1 0 0 1 0 0 J =   . (E.41) 1 0 0 0    0 0 0 1      Fisher Information Matrix for a Single Love Wave, Transla- tional Measurements The displacements induced by a single Love wave are

u (p, t) = α sin ψ cos(ωt κTp + ϕ) (E.42) x − − u (p, t) = α cos ψ cos(ωt κTp + ϕ) (E.43) y − uz(p, t) = 0 . (E.44)

The derivatives of the displacements with respect to the waveﬁeld parameters are ∂u x = sin ψ cos Φ (E.45) ∂α − n ∂u y = cos ψ cos Φ (E.46) ∂α n ∂u z = 0 , (E.47) ∂α

∂u ∂Φ x = α cos ψ cos Φ sin ψ sin Φ n (E.48) ∂ψ − n − n ∂ψ ∂u ∂Φ y = α sin ψ cos Φ cos ψ sin Φ n (E.49) ∂ψ − n − n ∂ψ ∂u z = 0 , (E.50) ∂ψ

∂u ∂Φ x = α sin ψ sin Φ n (E.51) ∂ϕ n ∂ϕ ∂u ∂Φ y = α cos ψ sin Φ n (E.52) ∂ϕ − n ∂ϕ ∂u z = 0 , (E.53) ∂ϕ 211 and ∂u ∂Φ x = α sin ψ sin Φ n (E.54) ∂κ n ∂κ ∂u ∂Φ y = α cos ψ sin Φ n (E.55) ∂κ − n ∂κ ∂u z = 0 . (E.56) ∂κ The elements of the FIM are obtained using (E.10). The diagonal elements are

N K = s (E.57) Iα,α 2σ2 α2N K = s (E.58) Iϕ,ϕ 2σ2 Ns α2K ∂Φ 2 = n (E.59) Iκ,κ 2σ2 ∂κ n=1 X Ns α2K ∂Φ 2 = N + n . (E.60) Iψ,ψ 2σ2 s ∂ψ n=1 ! X The oﬀ-diagonal elements are

= 0 (E.61) Iα,ϕ = 0 (E.62) Iα,κ = 0 (E.63) Iα,ψ Ns α2K ∂Φ = n (E.64) Iϕ,κ 2σ2 ∂κ n=1 X Ns α2K ∂Φ = n (E.65) Iϕ,ψ 2σ2 ∂ψ n=1 X Ns α2K ∂Φ ∂Φ = n n . (E.66) Iκ,ψ 2σ2 ∂ψ ∂κ n=1 X The FIM is

0 0 0 Iα,α 0 ϕ,ϕ ϕ,κ ϕ,ψ I(θ) =  I I I  . (E.67) 0 ϕ,κ κ,κ κ,ψ  I I I   0   ϕ,ψ κ,ψ ψ,ψ   I I I  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) 1 ω (p, t) = ακ sin(ωt κTp + ϕ) . (E.70) z 2 − The derivatives of the rotations with respect to the waveﬁeld parameters are

∂ω 1 z = κ sin(ωt κTp + ϕ) (E.71) ∂α 2 − ∂ω 1 z = ακ cos(ωt κTp + ϕ) (E.72) ∂ϕ 2 − ∂ω 1 1 ∂Φ z = α sin(ωt κTp + ϕ) + ακ cos(ωt κTp + ϕ) n (E.73) ∂κ 2 − 2 − ∂κ ∂ω 1 ∂Φ z = ακ cos(ωt κTp + ϕ) n . (E.74) ∂ψ 2 − ∂ψ

The elements of the FIM are obtained using (E.10). The elements on the diagonal are

κ2KN = s (E.75) Iα,α 8σ2 α2κ2KN = s (E.76) Iϕ,ϕ 8σ2 Ns α2K ∂Φ 2 = N + κ2 n (E.77) Iκ,κ 8σ2 s ∂κ n=1 ! X Ns α2κ2K ∂Φ = n ,2 (E.78) Iψ,ψ 8σ2 ∂ψ n=1 X and the oﬀ-diagonal elements are

= 0 (E.79) Iα,ϕ ακKN = s (E.80) Iα,κ 8σ2 = 0 (E.81) Iα,ψ Ns α2κ2K ∂Φ = n (E.82) Iϕ,κ 8σ2 ∂κ n=1 X Ns α2κ2K ∂Φ = n (E.83) Iϕ,ψ 8σ2 ∂ψ n=1 X Ns α2κ2K ∂Φ ∂Φ = n n . (E.84) Iκ,ψ 8σ2 ∂ψ ∂κ n=1 X The FIM is 213

0 0 Iα,α Iα,κ 0 ϕ,ϕ ϕ,κ ϕ,ψ I(θ) =  I I I  . (E.85) α,κ ϕ,κ κ,κ ψ,κ  I I I I   0   ϕ,ψ ψ,κ ψ,ψ   I I I  Comparing (E.85) and (E.67), we observe that the amplitude is now coupled with the wavenumber.

Fisher Information Matrix for a Single Rayleigh Wave, Trans- lational Measurements The displacements induced by a single Rayleigh wave are

u (p, t) =α cos ψ sin ξ cos(ωt κTp + ϕ) (E.86) x − T uy(p, t) =α sin ψ sin ξ cos(ωt κ p + ϕ) (E.87) − π u (p, t) =α cos ξ cos(ωt κTp + + ϕ) . (E.88) z − 2 The derivatives of the displacements with respect to the waveﬁeld parameters are

∂u x = cos ψ sin ξ cos(ωt + Φ ) (E.89) ∂α k n ∂u y = sin ψ sin ξ cos(ωt + Φ ) (E.90) ∂α k n ∂u π z = cos ξ cos(ωt + Φ + ) , (E.91) ∂α k n 2

∂u ∂Φ x = α sin ξ sin ψ cos Φ cos ψ sin Φ n (E.92) ∂ψ − n − n ∂ψ ∂u ∂Φ y = α sin ξ cos ψ cos Φ sin ψ sin Φ n (E.93) ∂ψ n − n ∂ψ ∂u π ∂Φ z = α cos ξ sin Φ + n , (E.94) ∂ψ − n 2 ∂ψ

∂u x = α cos ψ cos ξ cos Φ (E.95) ∂ξ n ∂u y = α sin ψ cos ξ cos Φ (E.96) ∂ξ n ∂u π z = α sin ξ cos(Φ + ) , (E.97) ∂ξ − n 2 214

∂u x = α cos ψ sin ξ sin Φ (E.98) ∂ϕ − n ∂u y = α sin ψ sin ξ sin Φ (E.99) ∂ϕ − n ∂u π z = α cos ξ sin(Φ + ) , (E.100) ∂ϕ − n 2 and ∂u ∂Φ x = α cos ψ sin ξ sin Φ n (E.101) ∂κ − n ∂κ ∂u ∂Φ y = α sin ψ sin ξ sin Φ n (E.102) ∂κ − n ∂κ ∂u π ∂Φ z = α cos ξ sin(Φ + ) n . (E.103) ∂κ − n 2 ∂κ The elements of the FIM are obtained using (E.10). The elements on the diagonal are N K = s (E.104) Iα,α 2σ2 α2KN = s (E.105) Iξ,ξ 2σ2 Ns Kα2 ∂Φ 2 = N sin2 ξ + n (E.106) Iψ,ψ 2σ2 s ∂ψ n=1 ! X NKα2 = (E.107) Iϕ,ϕ 2σ2 Ns Kα2 ∂Φ 2 = n . (E.108) Iκ,κ 2σ2 ∂κ n=1 X Oﬀ-diagonal elements are = 0 (E.109) Iα,ϕ = 0 (E.110) Iξ,ϕ = 0 (E.111) Iξ,κ = 0 (E.112) Iα,ψ = 0 (E.113) Iα,ξ = 0 (E.114) Iα,κ = 0 (E.115) Iψ,ξ Ns Kα2 ∂Φ = n (E.116) Iψ,ϕ 2σ2 ∂ψ n=1 X Ns Kα2 ∂Φ ∂Φ = n n (E.117) Iψ,κ 2σ2 ∂κ ∂ψ n=1 X Ns Kα2 ∂Φ = n . (E.118) Iκ,ϕ 2σ2 ∂κ n=1 X 215

The FIM is 0 0 0 0 Iα,α 0 0  Iϕ,ϕ Iϕ,κ Iϕ,ψ  = 0 0 . (E.119)  Iϕ,κ Iκ,κ Iψ,κ     0 ϕ,ψ ψ,κ ψ,ψ 0   I I I   0 0 0 0 ξ,ξ   I    Similarly to (E.67), also in E.119 the amplitude is decoupled from the other waveﬁeld parameters. In addition, also the ellipticity is decoupled from the other waveﬁeld parameters.

Fisher Information Matrix for a Single Rayleigh Wave, Rota- tional Measurements The rotations induced by a single Rayleigh wave are

ω (p, t) =ακ sin ψ cos ξ cos(ωt κTp + ϕ) (E.120) x − ω (p, t) = ακ cos ψ cos ξ cos(ωt κTp + ϕ) (E.121) x − − ωx(p, t) =0 (E.122)

∂ω x = κ sin ψ cos ξ cos(ωt + Φ ) (E.123) ∂α k n ∂ω y = κ cos ψ cos ξ cos(ωt + Φ ) (E.124) ∂α − k n ∂ω z = 0 , (E.125) ∂α

∂ω x = ακ sin ψ cos ξ sin(ωt + Φ ) (E.126) ∂ϕ − k n ∂ω y = ακ cos ψ cos ξ sin(ωt + Φ ) (E.127) ∂ϕ k n ∂ω z = 0 , (E.128) ∂ϕ

∂ω x = ακ sin ψ sin ξ cos(ωt + Φ ) (E.129) ∂ξ − k n ∂ω y = ακ cos ψ sin ξ cos(ωt + Φ ) (E.130) ∂ξ k n ∂ω z = 0 , (E.131) ∂ξ 216

∂ω ∂Φ x = α sin ψ cos ξ cos(ωt + Φ ) κ sin(ωt + Φ ) n (E.132) ∂κ k n − k n ∂κ ∂ω ∂Φ y = α cos ψ cos ξ cos(ωt + Φ ) + κ sin(ωt + Φ ) n (E.133) ∂κ − k n k n ∂κ ∂ω z = 0 , (E.134) ∂κ and

∂ω ∂Φ x = ακ cos ξ cos ψ cos(ωt + Φ ) sin ψ sin(ωt + Φ ) n (E.135) ∂ψ k n − k n ∂ψ ∂ω ∂Φ y = ακ cos ξ sin ψ cos(ωt + Φ ) + cos ψ sin(ωt + Φ ) n (E.136) ∂ψ k n k n ∂ψ ∂ω z = 0 . (E.137) ∂ψ

The elements of the FIM are obtained using (E.10). The elements on the diagonal are

κ2 cos2 ξN K = s (E.138) Iα,α 2σ2 α2κ2 cos2 ξN K = s (E.139) Iϕ,ϕ 2σ2 α2κ2 sin2 ξN K = s (E.140) Iξ,ξ 2σ2 Ns α2 cos2 ξK ∂Φ 2 = N + κ2 n (E.141) Iκ,κ 2σ2 s ∂κ n=1 ! X Ns α2κ2 cos2 ξK ∂Φ 2 = N + n . (E.142) Iψ,ψ 2σ2 s ∂ψ n=1 ! X The oﬀ-diagonal elements are 217

= 0 (E.143) Iα,ϕ ακ2 sin ξ cos ξN K = − s (E.144) Iα,ξ 2σ2 ακ cos2 ξN K = s (E.145) Iα,κ 2σ2 = 0 (E.146) Iα,ψ = 0 (E.147) Iϕ,ξ Ns α2κ2 cos2 ξK ∂Φ = n (E.148) Iϕ,κ 2σ2 ∂κ n=1 X Ns α2κ2 cos2 ξK ∂Φ = n (E.149) Iϕ,ψ 2σ2 ∂ψ n=1 X α2κ sin ξ cos ξN K = − s (E.150) Iξ,κ 2σ2 = 0 (E.151) Iξ,ψ Ns α2κ2 cos2 ξK ∂Φ ∂Φ = n n . (E.152) Iκ,ψ 2σ2 ∂ψ ∂κ n=1 X The FIM is

0 0 Iα,α Iα,κ Iα,ξ 0 0  Iϕ,ϕ Iϕ,κ Iϕ,ψ  I(θ) = . (E.153)  Iα,κ Iϕ,κ Iκ,κ Iκ,ψ Iξ,κ     0 ϕ,ψ κ,ψ ψ,ψ 0   I I I   0 0   Iα,ξ Iξ,κ Iξ,ξ    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 FIMs are additive.

I(θ) = It(θ) + Ir(θ) , (E.154) where It and Ir are the FIMs for the translational and rotational measurements, respectively.

E.3 Derivation of Cramér-Rao Bounds

The Cramér-Rao bound (CRB) is a lower bound on the variance of unbiased estimators Cramér (1946); Rao (1945). Knowledge of a lower bound on the 218 estimator variance has at least two practical implications. First, it allows us to evaluate the performance of an estimation algorithm, by enabling a quantitative comparison between the mean-squared estimation error (MSEE) of the algorithm under test and the smallest achievable variance. Second, the analytic expression of the CRB enables us to design the experiment set up in order to reduce the lower bound and therefore increase the amount of information gathered by the experiment. The information inequality states that the MSEE of an unbiased estimator is lower bounded as

T E θˆ E θˆ θˆ E θˆ (I(θ)) −1 . (E.155) − { } − { } where A B means that the matrix A B is PSD. In particular, we are interested in the diagonal elements of I−1 −as they provide a lower bound on the MSEEs of the corresponding parameters. Analytical inversion of the FIM derived in the previous sections is a tedious task. A possible approach is to invert the matrix numerically. This approach however, gives no insights of the dependency of the CRB on the parameters. To circumvent this limitation, in the following section we obtain analytical expression of the CRB for the parameters of interest through the equivalent Fisher information (EFI) .

Equivalent Fisher Information Since we are interested in the elements on the main diagonal of I−1 corresponding to wavenumber and ellipticity angle, we avoid the complete inversion of I as follows. We partition the FIM as

c dT I(θ) = , (E.156) d G ! where c is a scalar, d is a vector, and D is a matrix of suitable sizes. The element in the ﬁrst position of I−1 is then found using the Woodbury matrix identity to be

−1 (I(θ)) −1 = c dTG−1d , (E.157) 1,1 − 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 been referred to by some authors as− EFI Shen & Win (2010). In contrast with FI, the EFI accounts for the uncertainty introduced by the other unknown parameters of the statistical model. The term c is exactly the FI of the parameter of interest. The term dTG−1d is non-negative since G is PSD being a diagonal sub-block of a PSD matrix. This last quantity accounts for the uncertainty due to the other parameters. It is now clear that reducing the CRB is equivalent to increase the EFI. In other words, increasing the EFI is desirable as better estimation accuracy can be achieved. 219

In order to use (E.157) eﬀectively, it may be necessary to permute the row and columns 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′ = PTIP . In the following, we restrict ourselves to the analysis of the CRB of wavenumber 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 of the array in the coordinate system (a, b) instead of (x, y). The two coordinate systems are related as

a cos ψ sin ψ x = , (E.158) b sin ψ cos ψ y ! − ! ! where the angle or rotation is the azimuth ψ. Therefore a is the axis along the direction of propagation of the wave and b the axis perpendicular to it. In this rotated coordinate system we consider the new sensor positions (a , b ) introduce the center of gravity, or phase center, or the array { n n }n=1,...,Ns

Ns 1 a¯ = a (E.159) N n s n=1 X Ns 1 ¯b = b . (E.160) N n s n=1 X The following quantities are called moment of inertia of the array

Ns Q = (a a¯)2 (E.161) aa n − n=1 X Ns Q = (b ¯b)2 (E.162) bb n − n=1 X Ns Q = (a a¯)(b ¯b) , (E.163) ab n − n − n=1 X where the sensor are associated with unitary mass. An important remark is that the moment of inertia are invariant to a translation of the array. We observe the the moment of inertias (MOIs) are related to certain quantities that appear as elements of the FIMs. Recall the definition Φ = ωt n k − 220 x κ cos ψ y κ sin ψ + ϕ, we find observe that n − n Ns ∂Φ n = κ¯b (E.164) ∂ψ − n=1 X Ns ∂Φ n = N a¯ (E.165) ∂κ − s n=1 X Ns Ns ∂Φ ∂Φ n n = a b (E.166) ∂ψ ∂κ n n n=1 n=1 X X Ns ∂Φ 2 n = κ2 Q + N ¯b2 (E.167) ∂ψ bb s n=1 X Ns ∂Φ 2 n = Q + N a¯2 . (E.168) ∂κ aa s n=1 X Cramér-Rao Bounds Expressions We permute the rows/columns of the matrix as explained in (E.39) and then use (E.157) to obtain the following expressions. Further discussion on these expressions can be found in Maranò & Fäh (2013). We define 2 2 Ct = α K/2σt (E.169) 2 2 Cr = α K/2σr , (E.170) and 2 2 Φ = Ct + Crκ cos ξ (E.171) 2 2 2 Ψ = Ct sin ξ + Crκ cos ξ . (E.172)

Love Wave The MSEE of Love wave wavenumber, for translational measurements, is lower bounded as 2σ2 Q2 −1 E (ˆκ E κˆ )2 t Q ab . (E.173) − { } ≥ α2K aa − Qκ + N / 2 bb s The MSEE of Love wave wavenumber, for rotational measurements, is lower bounded as 8σ2 Q2 −1 E (ˆκ E κˆ )2 r Q ab . (E.174) − { } ≥ α2κ2K aa − Q bb The MSEE of Love wave wavenumber, for joint translational and rotational, is lower bounded as C C N /4 E (ˆκ E κˆ )2 C + κ2C /4 Q + t r s − { } ≥ t r aa C + C κ2/4 t r −1 2 2 2 Qab Ct + κ Cr/4 2 2 . (E.175) −C( tNs/κ + Ct + κ Cr/4)! 221

Rayleigh Wave The MSEE of Rayleigh wave wavenumber, for translational measurements, is lower bounded as

2 2 −1 2 2σt Qab E (ˆκ E κˆ ) Qaa . (E.176) − { } ≥ α2K − Q + N sin2 ξ/κ2 bb s The MSEE of Rayleigh wave ellipticity angle, for translational measurements, is lower bounded as

2 E ˆ E ˆ 2 2σt (ξ ξ ) 2 . (E.177) − { } ≥ α KNs n o The MSEE of Rayleigh wave wavenumber, for rotational measurements, is lower bounded as

2σ2 Q2 −1 E (ˆκ E κˆ )2 r Q ab . (E.178) − { } ≥ α2κ2 cos2(ξ)K aa − Qκ + N / 2 bb s The MSEE of Rayleigh wave wavenumber, for joint translational and rotational measurements, is lower bounded as

C C N cos2 ξ E (ˆκ E κˆ )2 ΦQ + t r s − { } ≥ aa C + C κ2 t r Q2 Φ2 −1 ab . (E.179) −CN sin2 ξ /κ2 + C cos2 ξN + ΦQ t s r s bb The MSEE of Rayleigh wave ellipticity angle, for joint translational and rotational measurements, is lower bounded as

2 E (ξˆ− E{ξˆ}) ≥ CtNs

n o − 2 2 2 1 κ Φ QaaΨNs − κ (Q − QaaQbb)Φ + ab . 2 2 2 2 2 3 NsCrCt cos ξ(ΨNs + κ ΦQbb) + QaaΨΦ Ns − κ (Qab − QaaQbb)Φ ! (E.180) Bibliography

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I would like to express my sincere gratitude to my advisor Prof. Donat Fäh for his supervision during the development of this thesis. His guidance and his enduring support truly helped me throughout the development of this work. I am truly thankful to Prof. Hans-Andrea Loeliger for co-advising this thesis. I learned a lot from his carefully thought advice. His comments were always hitting the essence of problems with a surgical precision. I thank Prof. Domenico Giardini for making my PhD possible and for accepting 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. I have fond memories of the weeks spent in Cambridge. I wish to thank Prof. Heiner Igel for serving as the external examiner of this thesis. I also wish to thank Prof. Johan Robertsson for accepting to chair the thesis committee. I thank my colleagues both at the Swiss Seismological Service and the Signal and Information Processing Laboratory for making the work days productive and enjoyable. Above everyone I would like to thank Dr. Christoph Reller with whom I shared a lot of my time thinking about this project. His input and eﬀort was very much appreciated.

230 About the Author

Short Biography

Stefano Maranò received the B.Sc. and M.Sc. degrees in Telecommunications Engineering from the University of Trento, Italy, in 2005 and 2008, respectively. From September 2007 to June 2008, he was with Laboratory for Information and Decision Systems at the Massachusetts Institute of Technology. Since 2009, he has been working towards a Ph.D. with the Swiss Seismological Service, ETH Zurich. In 2009, he was co-recipient of the Best Paper Award at the IEEE Global Communications Conference. His research interests lie in the broad area of signal processing. In particular the application of mathematical and statistical theories to challenging real- world problems.

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