Analysis of Rotational Motions induced by Earthquakes and Ocean Noise obtained by direct Ring laser measurements and array techniques.

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UNIVERSIT `A DI PISA Department of Earth Sciences

Msc. in Exploration and Applied Geophysics

Analysis of Rotational Motions induced

by Earthquakes and Ocean Noise

obtained by direct Ring laser

measurements and array techniques

by

Matteo Desiderio

Supervisor: Giorgio Carelli

Co-Supervisor: Andreino Simonelli

Examiner: Eusebio Stucchi

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UNIVERSIT `A DI PISA

Abstract

Msc. in Exploration and Applied Geophysics

by Matteo Desiderio

In the present work, rotational and translational ground motions are analyzed, in order to retrieve the local wavefield properties. The same method of analysis is applied to both earthquake and ambient noise signals, in order to estimate the wavefield direction and the phase velocity as a function of frequency. In the first application, rotations are directly measured by means of a Ring Laser Gyroscope (RLG), named GINGERino. In the second case, they are derived from a narrow-aperture array of seismometers (code-named XG), thanks to a finite difference scheme called Array Derived Rotation (ADR). In both cases, the method used is a 4 Components (4C) analysis, which combines the rotation rate around the vertical axis and the three components of translational ground motions. The location of both experiments is an underground gallery beneath mount Gran Sasso (Italy). A Python module is written to obtain the ADR and several Obspy routines (a Python library for observational seismology) are used.

A central theme of this thesis is therefore the joint analysis of rotational and transla-tional ground motions. The addition of rotations, both measured and derived, poses some significant advantages. First, in terms of logistics: thanks to these signals, wave-field properties can be estimated through single station measurements or narrow aperture arrays, whereas traditionally these are obtained through large and dense networks. Sec-ondly, in terms of information: 6C and 4C analyses allow for a deeper characterization of the wavefield. In the case of 6C, for example, rotations and translations around all axes can be analyzed and discriminated. As for the present work, the focus is on 4C, which is able to separate SH from P-SV: in fact, SH polarized plane waves are the only ones capable of exciting rotations with respect to a vertical axis. All of these points are expanded on in Chapter One, the Introduction.

The Second Chapter is dedicated to general background theory. Here, the mathematical relationship between rotational and translational ground motions is described and used as a base for the 4C analysis. This is the key to obtain the local back-azimuth (BAZ) and, more importantly, the local phase velocity (under some assumptions). The ADR procedure and its limits are then described, along with the f-k analysis technique: the

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latter is an array processing method also used for BAZ and apparent horizontal velocity estimation. Finally, some sections are devoted to the microseism: namely, how it is originated, the importance of source identification in geophysics and how P-body waves noise sources can be modeled.

In the Third Chapter, a 4C analysis is applied to earthquake data from November and December 2019. The aim is to estimate the local BAZ and analyze the phase velocity of the seismic waves as function of frequency: this represents a first step towards a study of the velocity profile under the Gran Sasso. The phase velocity values obtained show the normal dispersive behavior of surface waves. However, most strikingly, it appears that the BAZ is systematically underestimated with respect to the theoretical one: this suggests local lateral variations in the velocity structure.

In the Fourth Chapter, XG is employed to obtain the ADR and then use it in a 4C analysis. This is applied to the secondary microseism, a type of ocean noise. In order to validate such a workflow, tests are carried out in a first section. These are achieved by simulating data for a large teleseismic event as it would have been measured by XG. For the same network, a parallel test is also run for an f-k technique, since this is a common array processing method. The objective is to establish which one of the two methods is the most suitable for the available array: the result is that the XG is too narrow in order for the f-k to have success, in opposition to the ADR. In addition, this further demonstrates the advantage of 4C analysis, logistics-wise: in fact, deploying a larger array would have been impossible due to space constraints. In a later section, the microseism is finally examined. Two sources of noise with different BAZs and spectral content are found. For each of them, a dispersion analysis is attempted. Finally, theo-retical P body waves noise sources are computed and compared to the estimated BAZ: an accordance is found in the period band [1-4] seconds, suggesting a common source for SH and P noise-excited waves.

The final Fifth Chapter represents a general conclusion to this work, summing up all the results and outlining some other potential applications of rotational measurements, especially in the field of active seismology.

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List of Figures

2.1 Examples of two arrays for the f-k and their transfer functions. . . 14

3.1 Translational and rotational waveforms for Event n. 1. Traces are filtered in band [0.02, 5.00] Hz for visualization purposes. Theoretical P and S first arrivals are depicted. . . 23

3.6 Top panel: aT and Ωz for Event n.1. Bottom panel: C matrix, picked

maxima, theoretical and observed BAZ values. The bands used for each panel are shown. . . 26

3.11 Dispersion analysis for Event n.1. The central periods of each band are on the left-hand of the plot. On the right-hand of the plot, c estimates from peak-envelope rates are listed for each band.. . . 30

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List of Figures vii

4.1 Position and scheme of the XG network. Station IV.GIGS is also depicted. 37

4.2 PPSD at station GIGS, East Channel, from 31/12/2018 to 19/01/2019. . 37

4.3 Velocity traces for the simulated Tohoku event at the reference station. . 40

4.4 Time-frequency analysis for the simulated Tohoku event, with no filter-ing. First panel: ADR. Second and third panel: East and North derived acceleration components.. . . 41

4.5 Time-frequency analysis for the simulated Tohoku event, with pre-emptive filtering of translational traces. First panel: ADR. Second and third panel: East and North derived acceleration components. . . 41

4.6 Comparison between ADR and derived acceleration traces, after filter-ing. First panel: Ωz and aE. Second panel: Ωz and aN. Time axis is

concentrated on surface waves. . . 42

4.7 Comparison between ADR and derived acceleration traces, with no pre-emptive filtering. First panel: Ωz and aE. Second panel: Ωz and aN.

Time axis is concentrated on surface waves. . . 42

4.8 Top panel: aT, as derived from estimated BAZ, and ADR. Bottom panel:

C matrix and estimated BAZs as function of time; weighted average of points is also represented. . . 43

4.9 aT (based on estimated BAZ) values against ADR. The time window

chosen for this plot is [2600, 3215] s. . . 44

4.10 Map of theoretical epicentre and estimated BAZ obtained from 4C analysis. 44

4.11 ASDs of ADR and aT as obtained from estimated BAZ. The bands for

the dispersion analysis are also depicted. . . 45

4.12 Dispersion analysis for simulated Tohoku event. The central periods of each band are on the left-hand of the plot. On right-hand of the plot, c estimates are listed for each band. . . 46

4.13 Relative beam power for XG array, as function of misfit between true and estimated wave-number components. . . 48

4.14 Result of f-k analysis for the simulated Tohoku event: BAZ at each time-step, from optimal apparent horizontal slowness components. Relative beam power is indicated. . . 48

4.15 Result of f-k analysis for simulated Tohoku event: apparent horizontal velocity module at each time-step, from estimated/optimal slowness com-ponents. Relative beam power is shown. . . 49

4.16 Comparison between ADR and aT, as derived from 4C-estimated BAZ.

Each row corresponds to a noise level. Left side: ASDs, with grayed-out area representing filter band. Right side: waveforms within surface-wave time-window. . . 51

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List of Figures viii

4.18 Plot of residuals between estimated and theoretical BAZ. Top panel: 4C analysis. Bottom panel: f-k analysis. . . 54

4.19 Residuals between results of the two methods of BAZ estimate, f-k and 4C. Errorbars are given by propagating the uncertainties of the two methods. 54

4.20 Histogram of correlation coefficients between simulated aT ,thand 300

real-izations of the ADR, obtained by perturbing the coordinates of the virtual receivers. . . 55

4.21 Map of relative beam power as a function of horizontal slowness compo-nents. The result is referred to the last time-step. . . 56

4.22 Velocity traces recorded at all four stations of the XG array, from 16:00 of 09/01/2019 to 15:00 of 10/01/2019. Note that a different scale is used for each of the four plots. . . 59

4.23 Amplitude spectral density of the signals at each channel for each station of the XG Network. . . 60

4.24 ADR and acceleration traces at the reference station for the [1.25, 10] s band microseism. First panel: Ωz and aE. Second panel: Ωz and aN. A

zoomed in section is represented in the detail. . . 61

4.25 Zoomed-in ADR and acceleration traces at the reference station for the [1.25, 10] s band microseism. . . 62

4.26 Comparison between ASDs of ADR and aE, aN at the reference station

for the microseism. . . 62

4.27 Time-frequency analysis of the microseism. First panel: Ωz. Second and

third panel: aE and aN. . . 63

4.28 Matrix of 0-lag correlation coefficients C for the [1.25, 10] s band microseism. 64

4.29 Hour by hour estimate of the BAZ for the [1.25, 10] s band microseism. The weighted average of the points is also represented. . . 65

4.30 ADR waveform and ASD for the [1.25, 4] s microseism. . . 66

4.31 Matrix of 0-lag correlation coefficients C for the [1.25, 4] s band microseism. 66

4.32 Hour by hour estimate of the BAZ for the [1.25, 4] s band microseism. The weighted average of the points is also represented. . . 67

4.33 ADR waveform and ASD for the [4, 10] s microseism. . . 68

4.34 Matrix of 0-lag correlation coefficients C for the [4, 10] s band microseism. 68

4.35 Hour by hour estimate of the BAZ for the [4, 10] s band microseism. The weighted average of the points is also represented. . . 69

4.36 Time evolution of two PPSD frequency bins at the GIGS station for the microseism. These are the bins closest to [1.25, 4] s and [4, 10] s bands. . 69

4.37 Comprehensive plot of hour-by-hour BAZ estimates for the microseism in all bands. Averages are also represented. . . 70

4.38 Time-averaged and smoothed correlation values above fixed threshold. Each panel corresponds to the reported threshold and band. . . 71

4.39 Top panel: aT, as derived from estimated BAZ, and ADR in the [1.25,

10] s band; the detail represents a zoom-in. Bottom panel: C matrix and hourly BAZs estimate; weighted average of points is also represented.. . . 72

4.40 Top two panels: aT and ADR; C matrix and hourly BAZs estimate in the

[1.25, 10] s band. Bottom two panels: same as first two panels, but in [4, 10] s. . . 73

4.41 Compared ASDs of aT and ADR. The bands used for the dispersion

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List of Figures ix

4.42 Dispersion analysis for the microseism. The angle used to derive aT is 129◦. 75

4.43 Dispersion analysis for the microseism. The angle used to derive aT is 329◦. 75

4.44 Comparisons between BAZs of theoretical P sources at varying seismic periods and observed BAZs Z in the band [1.25, 4] s. . . 77

4.45 Comparisons between BAZs of theoretical P sources at varying seismic periods and observed BAZs Z in the band [4, 10] s. . . 77

4.46 Comparison between Z in the band [1.25, 4] s and BAZs of theoretical P sources at corresponding period. . . 78

4.47 Comparison between Z in the band [4, 10] s and BAZs of theoretical P sources at corresponding period. . . 78

A.1 BAZs of the microseism estimated from the f-k analysis in the two fre-quency bands, compared to the 4C analysis results. . . 85

B.1 Scheme of an array designed for an f-k analysis in the wavelength range of [10, 40] km.. . . 87

B.2 Relative beam power for the ideal array, as function of misfit between true and estimated wave-number components. . . 88

B.3 Scheme of an array designed for the ADR for wavefields with wavelength longer than 6.4 km.. . . 89

B.4 Time frequency analysis for the simulated Tohoku event, as measured by an ideal array. First panel: ADR. Second and third panel: horizontal acceleration components. The translational traces are filtered before the ADR. . . 90

B.5 Comparison between derived acceleration traces and ADR for an ideal array. First panel: Ωz and aE. Second panel: Ωz and aN. Traces are

filtered before the ADR. . . 90

B.6 Top panel: aT, as derived from estimated BAZ, and ADR of an ideal

array. Bottom panel: C matrix and estimated BAZs as function of time; weighted average of points is also represented. . . 91

B.7 Map of theoretical epicentre and estimated BAZ obtained from 4C anal-ysis for the ADR-ideal array. . . 91

B.8 Result of f-k analysis for simulated ideal array: errorbars in the usual surface-waves time-window are depicted. Relative beam power is shown. . 92

B.9 Relative beam power map as a function of horizontal slowness compo-nents, for the ideal array. The result is referred to the last time-step. The -3 dB contour for uncertainty estimation is shown. . . 92

B.10 Comparison between ADR and aT, as derived from 4C-estimated BAZ,

through an ideal array. Each row corresponds to a noise level. Left: ASDs, with filter band represented. Right: waveforms in surface-wave time-window. . . 94

B.11 Comparison between theoretical BAZs and results of f-k and 4C analysis from ideal arrays. . . 95

B.12 Plot of residuals between theoretical BAZs and estimates from ideal ar-rays. Top panel: 4C analysis. Bottom panel: f-k analysis. . . 96

B.13 Residuals between results of the two method of BAZ estimate, f-k and 4C, as obtained by their respective ideal arrays. Errorbars are obtained from uncertainty propagation. . . 96

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List of Tables

3.1 Specifics of the 4C station employed. . . 21

3.2 List of numbered earthquake events analyzed, in chronological order. . . . 22

3.3 Summary of the results obtained for the BAZ analysis, along with fre-quency band used. . . 29

4.1 Coordinates of the array’s stations. The nominal elevation is 960 m across the whole array. The precision for each position is given by the least significant digit, once it is converted from degrees to meters.. . . 38

4.2 Technical specifications of the seismometers used for the XG array. . . . 38

4.3 Technical specifications of the digital recorders used for the stations of the XG array. . . 38

4.4 BAZ estimation results for simulated XG array at increasing noise levels, for f-k analysis (left side) and 4C analysis (right side). . . 50

4.5 Final estimates of back-azimuth, as a result of weighted average of the hourly estimate and temporal mean of ZLCC values above threshold. . . . 70

B.1 BAZ estimation results for simulated ideal arrays at increasing noise lev-els, for f-k analysis (left side) and 4C analysis (right side). This time, a correct method is used to estimate the uncertainty in the f-k. . . 93

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Abbreviations

4C Four Components

6C Six Components

ADR Array Derived Rotation

ASD Amplitude Spectral Density

ZLCC Correlation Coefficient

BAZ Back-azimuth

CMT Centroid Moment Tensor

GINGER Gyroscopes in General Relativity

IFREMER Institut Fran¸cais de Recherche pour l’Exploitation de la Mer INFN Istituto Nazionale di Fisica Nucleare

INGV Istituto Nazionale di Geofisica e Vulcanologia

IOWAGA Integrated Ocean Waves for Geophysical and Other Application IRIS Incorporated Research Institutions for Seismology

LNGS Laboratori Nazionali del Gran Sasso

MAD Median Absolute Deviation

NHNM New High Noise Model

NLNM New Low Noise Model

NTP Network Timing Protocol

PPSD Probabilistic Power Spectral Density

PREM Preliminary Earth Model

RLG Ring Laser Gyroscope

RMS Root Mean Ssquare

SNR Signal to Noise Ratio

ZLCC Zero-lag Correlation Coefficient

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Chapter 1

Introduction

Recent developments in the instrumentation, with sensors capable of measuring rotation rates with a sensitivity as high as 10−2 nrad/s/√Hz [1], have propelled an advancement in the field of rotational seismology. Although portable sensors suitable for seismic measurements (like the BlueSeis-3A) are already available and employed [2], the most sensitive instruments are still large Ring Laser Gyroscopes (RLGs).

RLGs have been used in conjunction with seismometers for 4 Components (4C) or 6 Components (6C) analyses of a variety of signals, from local earthquakes [3] to ocean noise [4]. Nevertheless, arrays of seismometers have also been employed to approximate ground rotational motions at the surface [5], [6]. It has been shown [7] that an accurate Array Derived Rotation (ADR) is obtained with respect to the true rotation rate, with a better Signal to Noise Ratio (SNR) when a high number of stations is used. On the other hand, the array’s extension D must be narrower than λ/4 (λ denotes the dominant wavelength of the signal) in order for the ADR to be accurate [5].

Nevertheless, the addition of rotational ground motions to the analysis entails signifi-cant advantages. First of all, rotational measures by RLGs are not contaminated by translations, whereas seismometers are sensitive to tilting [1].

More importantly, a common objective is the extraction of wavefield properties such as direction of propagation and phase velocity. Traditionally, this is achieved through techniques (e.g. f-k analysis) that rely on the delays measured across the stations of dense and wide networks. Single-station 4C/6C analyses are able to achieve the same result but present a clear logistical advantage whenever large arrays cannot be deployed. This may happen in a variety of scenarios: when too few instruments are available (e.g. ocean-bottom and extraterrestrial seismology) or in areas with limited access (e.g. urban settings, mines, boreholes) [8]. To some extent, this also applies when the rotation rate is array-derived. In fact, for a given wavelength range, the ADR requires D < λmin/4,

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Introduction 2

while, for a common f-k analysis, D > λmax [9]. Hence, arrays for 4C/6C analysis are

potentially more compact.

However, the most significant advantage is from an information standpoint. It can be shown [8] that a 6C polarization analysis is able to:

• Unambiguously separate the wavefield into all different seismic phases (Surface Love and Rayleigh waves, body P, SV, SH waves).

• Extract the back-azimuth and true incidence angle for each of them. • Estimate the local shear and bulk velocities.

This is possible even for multiple overlapping arrivals. This entails some useful applica-tions to active seismology, that are outlined in Chapter5.

A 4C analysis, such as the one discussed in this thesis, is also powerful: since only SH polarized waves generate vertical rotations [8], it is possible to separate P-SV phases from SH ones. This allows to apply the theory of rotations to Love waves and obtain their BAZ and phase velocity [1].

There is a wide array of applications for 6C and 4C measurements (e.g. seismic source inversion, seismometer tilt correction, exploration seismology). As far as this thesis is concerned, two main topics are addressed: site characterization and ambient noise study. In the first case, vertical rotation rate and transverse acceleration from earthquakes are compared in order to estimate phase velocity as a function of frequency. This study is meant to add additional observations to the ones made by [3], with the perspective of building a large dataset suitable for the characterization of the Gran Sasso site. In fact, dispersion curves can be inverted to obtain velocity profiles: this is shown in [10], where this workflow is presented as a method for microzonation, another possible application of rotational seismology.

In the second case, the 4C analysis capability of estimating the BAZ is ideal to address the problem of non-diffusivity of sources in ambient noise tomography. In fact, this method relies on the assumption of a homogeneous distribution of noise sources [11], [12]: however, ocean noise is highly anisotropic, with repercussions on the accuracy of the result. BAZ extraction is necessary to estimate such an error, and the 4C analysis is presented as a suitable alternative to other techniques for BAZ retrieval (e.g. f-k). A secondary objective is a dispersion analysis applied to ambient noise data, in a similar fashion as described in the previous paragraph. Finally, theoretical noise sources are compared to the observed ones. Despite the fact that SH waves cannot propagate in water, a significant part of the microseism consists of Love waves [4]. A mechanism of phase conversion could be then invoked as an explanation: as a basis for a study in that

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Introduction 3

direction, theoretical P body waves sources are computed, so that they can be compared to the BAZs observed for SH waves.

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Chapter 2

Background Theory

The aim of this chapter is to provide a theoretical framework useful for later discussion. In a first section, theory of rotations is outlined, since it is a basis for the 4C data analysis techniques used throughout this work. These are described in a dedicated section. Then, a section is devoted to array processing methods: here, the ADR and the f-k method of analysis are described. The first one is used to estimate the ground rotation for later use in a 4C analysis; the latter is directly aimed at BAZ and apparent horizontal velocity estimation. Finally, a section is devoted to the microseism or ocean ambient noise: its origin and value in geophysics are outlined, along with a model for body waves generation.

2.1 Rotational and Translational Ground Motions

Let {ˆx1, ˆx2, ˆx3} represent a right-handed orthogonal coordinate system. As theory

pre-dicts [13], rotations can be expressed as the curl of the displacement field ~u(x1, x2, x3)

:     θ1 θ2 θ3     = 1 2∇ ×     u1 u2 u3     (2.1)

Its time derivative gives the rotation rate ~Ω:

∂~θ ∂t = ~Ω = 1 2∇ ×     v1 v2 v3     = 1 2     ∂2v3− ∂3v2 ∂3v1− ∂1v3 ∂1v2− ∂2v1     (2.2)

where ~v(x1, x2, x3) is the velocity field and ∂i denotes the partial derivative with

re-spect to the i−th coordinate. Finally, it should be noted that, throughout this work, 4

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Background Theory 5

{ˆx1, ˆx2, ˆx3} will be considered aligned to East, North and vertical Z directions.

At this point, the case when the wavefield is generated by surface waves can be exam-ined. However, a brief theoretical overview is first given: plane waves will be assumed for simplicity’s sake. Surface waves originate when body waves with over-critical an-gles of incidence are met with certain boundary conditions that typically apply near the surface [14], [15]. When P-SV polarized waves are involved, Rayleigh surface waves are spawned, while SH phases give way to Love waves. In the first case, the boundary between free air and a homogeneous half-space is sufficient. In the latter case, a slightly more complicated setup is necessary: the simplest condition that can sustain Love waves is a wave-guide, that is, a homogeneous surface layer bearing over a homogeneous half-space with higher shear velocity.

Moreover, as far as polarization is concerned, Rayleigh waves have an elliptical polarisa-tion, with semi-axes aligned to two directions: longitudinal and vertical-transverse with respect to the propagation direction. Love waves, on the other hand, are SH-polarized. In addition, surface waves can be dispersive: this means that their phase velocity can be a function of frequency. Rayleigh waves are only dispersive when they propagate in an in-homogeneous or stratified half-space. Love waves, on the other hand, are always dispersive. Let ∆h be the depth of a wave-guide with shear velocity β1 and modulus µ1,

over a half-space with β2 > β1 and µ2. It can be shown [14] that the dispersion relation

for Love waves is:

tan 2πf ∆h q β₁−2− c−2_L = µ2 q c−2_L − β₂−2 µ1 q β−2₁ − c−2_L (2.3)

where f is the frequency and cL is the phase velocity (with β1 ≤ cL ≤ β2) [15], [14].

Phase velocity clearly depends on f , besides the other fixed physical parameters. Also, for a fixed f , a finite set of cL solutions can be found, called modes. For a given mode,

it can be shown [14] that, as f increases, cL becomes asymptotically equal to β1. On

the contrary, as f approaches 0, cL tends to β2: as a consequence, low-frequency

com-ponents of a Love wave are more ’sensitive’ to deeper layers of the Earth and thus travel at higher velocity1, as opposed to high frequencies.

Now, the theory of rotations is applied to surface waves.

Consider a transverse Love wave propagating horizontally, on the surface (x3 = 0) of

a laterally homogeneous medium. The symmetry of the medium allows to rotate the reference frame, so as to match ˆx1 with the propagation direction and ˆx2 with the

trans-verse axis of polarization ˆT . This causes no loss of generality. Under the assumption of

1

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Background Theory 6

a plane wave, the components of the displacement field at the surface are then:

u1 = 0, u2 ≡ uT = Aeiω(x1/cL−t), u3= 0 (2.4)

where ω = 2πf . By applying equation 2.1 to uT and recalling that ∂tuT = −cL∂xuT,

the following relationship is obtained [3], [1]:

Ωz ≡ Ω3 = −

1 2cL

aT (2.5)

This links the vertical rotation rate (denoted in the following as Ωz) to the transverse

acceleration aT at a given point on the surface and allows to obtain an estimate of cL,

if ground translations and rotations are available. It is worth noting that the other components of ~Ω are null. A similar derivation can be carried out for plane Rayleigh waves traveling along x1, resulting in

Ω2= −

1 cR

a3 (2.6)

Furthermore, it must be reminded that rotational motions are also produced by body phases, not just surface waves. In this case, discontinuities in the medium can have consequences: just as an example, P waves do not give way to rotations, unless they interfere with their reflections from an interface with the free surface [8]. Finally, it is worth noting that only SH polarized phases (either body or surface waves) originate a vertical rotation rate. This allows for a complete separation from P-SV phases when Ωz

is measured or estimated. However, equation 2.5is valid for Love waves only.

2.2 Data Processing Methods

This section is divided into two sub-sections. In the first one is a description of the 4C analysis, aimed at BAZ and phase velocity estimation and employed throughout this work. The required rotation rate can be either directly measured or array derived: the technique used to obtain the latter (ADR) is described in the second sub-section. Here, the f-k analysis is also outlined. This array method is also used to estimate BAZs and seismic velocities.

2.2.1 4C Analysis: Backazimuth and Dispersion Analysis

Based on2.5, if −aT and Ωz from a Love wave are measured at a site, they are expected

to be completely in phase. However the sensor’s channels may not be aligned to the transverse-radial axes: in this case, it is necessary to rotate the recorded acceleration

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Background Theory 7

traces to derive aT. If the translational sensor is aligned to the East and North

direc-tions, by considering the transverse-radial reference frame at the receiver, the following expression can be used:

aT(θ) = aEcosθ − aNsinθ (2.7)

where θ denotes the local BAZ of the wavefield.

However, in this problem, the angle is unknown. A grid-search can be thus performed to estimate it. For a given θ, the signals are divided in equal time-windows, beginning at instants t: within those, a statistical Zero-Lag Correlation Coefficient (ZLCC) is computed between −aT and Ωz. This yields a value C, ranging from -1 to +1:

C = h[haTi − aT][Ωz− hΩzi]i saTsΩz

(2.8)

Here, the h i operator denotes the mean of the samples considered in the time-window, while the symbol s represents the standard deviation. This is repeated for discrete values of θ, spanning from 0◦ to 360◦, with δθ-wide steps. This yields a matrix of values C(θ, t), which are expected to be similar to one for θ close to the real BAZ, for the duration of the seismic perturbation. A final BAZ estimate can be finally found by picking the BAZ values corresponding to the higher ZLCCs and averaging those in a convenient time interval. Alternatively, this can be also adapted to obtain the BAZ as a function of time. In later sections, both methods will be used.

Once the BAZ for the local wavefield is available, equation2.7can be used to obtain aT.

Then, based on equation2.5, an estimate for cLcan be given. Love waves are dispersive,

as explained earlier in this chapter: therefore the ratio between aT and Ωz is actually

frequency-dependent. The dispersion can be analyzed by filtering the two traces in 1₂ or 1₃-octave bands, centred at different frequency values fc. The envelopes of the two

signals are then computed and equation 2.5 is applied to their respective peak values: this gives cL values as a function of frequency fc.

As explained earlier in this chapter, Love waves at lower frequencies tend to sample deeper layers2. If dispersion curves are available from Love waves analysis, they can be inverted to obtain shear-velocity profiles at the site of the experiment. The dispersion analysis can be therefore thought of as a first step towards a study of local geological structure. However, this goes beyond the scope of the present work.

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Background Theory 8

2.2.2 Array Methods

A seismic array is an ensemble of synchronized seismic sensors, with common response function, deployed in order to sample a wavefield in space and time and infer some of its features. From an analysis point of view, arrays are characterized by the collective processing of waveforms [16], [9]. Geometrically, an array is characterized by its aperture D, that is its maximum horizontal extension, and d, which is the average spatial sampling interval. In this subsection, two array techniques are examined: the ADR and the f-k analysis.

2.2.2.1 Array Derived Rotation

From equation 2.2, it is apparent that if ~Ω is to be derived from translational ground motions, an estimate of the wavefield’s horizontal gradient is needed. This information is contained in the displacement gradient matrix J :

J =     ∂1u1 ∂2u1 ∂3u1 ∂1u2 ∂2u2 ∂3u2 ∂1u3 ∂2u3 ∂3u3     (2.9)

This matrix has 9 independent components. It can be decomposed in a symmetric and asymmetric part, which are, respectively, the strain and rotation tensors, and Θ. Let G denote the time-derivative:

˙

J = G (2.10)

In a real experiment, the wave field is measured, at discrete instants, by an array of N seismic sensors at different points in space. Let ~x(n)=

h

x₁(n) x₂(n)x₃(n) iT

denote the position, before any seismic perturbation, of the n−th seismograph, with n = 0, 1 . . . N − 1. The station at n = 0 is the reference station. Let also ~v(n)=

h

v₁(n) v₂(n) v₃(n) iT

be the seismic trace recorded at the n−th sensor, at every instant. At this point, the following vectors can be introduced:

~ ∆x =        ~ x(1)_{− ~}_x(0) ~ x(2)− ~x(0) .. . ~ x(N −1)− ~x(0)        =        ~ ∆x(1) ~ ∆x(2) .. . ~ ∆x(N −1)        (2.11)

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Background Theory 9 and ~ ∆v =        ~ v(1)− ~v(0) ~ v(2)− ~v(0) .. . ~v(N −1)− ~v(0)        (2.12)

For now, let N = 2, so that ~∆x = ~∆x(1), ~∆v = ~∆v(1). Under the approximation of a uniform gradient of the wave field, that is

∂vi

∂xj

≈ ∆vi ∆xj

(2.13)

it can be shown that [5]:

~

∆v = G ~∆x (2.14)

with Gij = _∂x∂vi_j. For a given wavelength λ, it can also be demonstrated that the

rela-tionship in 2.13 is more than 90% accurate if D < λ/4 [5], [6]. This can be considered as a pre-requisite for the ADR procedure.

Now, it should be reminded that the unknowns that must be derived are the compo-nents of G. Conveniently, under certain conditions, it is possible to reduce the number of independent components. To that aim, recall Hooke’s law for a homogeneous isotropic medium:

σij = λkkδij+ 2µij (2.15)

where λ and µ are the Lam`e constants. If the sensors are located near the surface, a stress-free boundary condition can be applied [7], [17], that is σi3 = 0 for i = 1, 2, 3.

Consequently, the number of independent components of J and G is reduced to 6:

G =     ∂1v1 ∂2v1 ∂3v1 ∂1v2 ∂2v2 ∂3v2 −∂3v1 −∂3v2 −η(∂1v1− ∂2v2)     (2.16)

with the dimension-less parameter η = _λ+2µλ .

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Background Theory 10 system:          ∆v1 = ∆x1p1+ ∆x2p2+ ∆x3p3 ∆v2 = ∆x1p4+ ∆x2p5+ ∆x3p6 ∆v3 = −∆x1p3− ∆x2p6− ∆x3η [p1+ p5] (2.17)

where the vector ~p is defined as:

~ p =             ∂1v1 ∂2v1 ∂3v1 ∂1v2 ∂2v2 ∂3v2             (2.18)

It is apparent that three more equations are needed to solve the system exactly: this can be attained by adding a third seismic station. Therefore, with N = 3:

~ ∆v = R~p = =             ∆x₁(1) ∆x₂(1) ∆x₃(1) 0 0 0 0 0 0 ∆x₁(1) ∆x₂(1) ∆x₃(1) −η∆x₃(1) 0 −∆x₁(1) 0 −η∆x₃(1) −∆x₂(1) ∆x₁(2) ∆x₂(2) ∆x₃(2) 0 0 0 0 0 0 ∆x₁(2) ∆x₂(2) ∆x₃(2) −η∆x₃(2) 0 −∆x₁(2) 0 −η∆x₃(2) −∆x₂(2)             ~ p (2.19)

where R is a forward-matrix. In fact, it is possible to obtain ~p by solving the corre-sponding exactly-determined inverse problem, in order to obtain the ground rotations about the reference station [17]. This system can be expanded by simply adding more seismometers, thus resulting in an over-determined inverse problem (R has more rows than columns): for this work, a python module was created to build R from geograph-ical coordinates stored in a dataless miniseed file and carry out the inversion with a least-square formula:

~

p = R−g∆v~ (2.20)

with the generalized inverse defined as:

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Background Theory 11

This is used in chapter 4 to obtain Ωz = 1₂(p4 − p2), but can be applied to obtain

all components of p. Finally, it is worth mentioning that if the stations share the same elevation x3, the problem of estimating η for the site of the experiment can be

conveniently disregarded.

2.2.2.2 ADR and Uncertainties

As stated in section2.2.2.1, a least-square formula is employed for the inversion in this work. However, a weighted least-square method3 can be also used if the data covariance matrix C∆v is available:

~

p =RTC_∆v−1R−1RTC_∆v−1∆v~ (2.22) This can be useful if some stations appear to be noisier than others: in fact, this way they can be inversely weighted by their variance. The C∆v matrix can be built upon

Cv, which is the covariance matrix of the seismograms for the whole array. This is a

3N × 3N diagonal matrix, storing the variances σ_i(n)2 of the noise at each channel i of each station n. It can be showed [17] that:

~ ∆v = D           v₁(0) v₂(0) v₃(0) .. . v(N −1)₃           (2.23)

where D is a 3N × 3N matrix. This is defined as:

D =        −I3 I3 03 . . . 03 −I₃ 03 I3 .. . . .. 03 −I₃ 03 03 I3        (2.24)

with I3 and 03 3 × 3 identity and null matrices respectively. Theory [18] allows to

com-pute C∆v = DCvDT. This is not a diagonal matrix, since the components of ~∆v consist

of differences with respect to the same reference station and, therefore, are correlated.

The data covariance matrix allows for a weighted inversion but, more importantly, can also be used to propagate the uncertainty on ~Ω. In fact the parameters covariance ma-trix is Cp= R−gC∆vR−gT. Hence, for example, the uncertainty4on the vertical rotation

3_{Or least χ}2_. 4

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rate is [18]:

σΩz =

1

2pCp22+ Cp44− 2Cp24 (2.25)

For all of this to be possible, the standard deviations for all velocity traces must be known. If the target is an earthquake-induced rotation rate, then those can be estimated from the Root Mean Square (RMS) of the signal before the onset of the seismic wave [7]. However, this method of estimate becomes challenging if the target is a continuous signal in time: this is the case of ocean ambient noise, for which the ADR is computed in chapter 4. For this reason, a deliberate choice not to propagate the uncertainty on microseism-induced rotations is made in the following discussion. On the other hand, an estimate on the uncertainty of ~∆v could be obtained from the misfit between measured and theoretical data points. However, this is not suitable for a χ2 test for the goodness of fit [20]. It could be then used to propagate, in some way, an uncertainty on the BAZ estimated from the 4C analysis: however, in chapter4, this will be obtained in an alternative way.

2.2.2.3 Beamforming and f-k Analysis

Now, the principles of the f-k analysis technique are briefly described. This can be used to obtain BAZ and horizontal velocity estimates, just like the 4C method in section

2.2.1. The description here mirrors the one found in [16], [9] and is limited to plane waves for simplicity.

Beamforming5 _{refers to a method wherein signals recorded at different stations are}

de-layed and summed to improve the SNR. The rationale is that waveforms across a group of stations only differ by some time delay and local noise (if spatial coherency of the wavefield across the array and negligible attenuation are assumed). By shifting in time the traces wn(t) at each station n by an optimal value τn, it is possible to maximize the

beam, defined as:

b(t) = 1 N N −1 X n=0 wn(t + τn) (2.26)

Assuming a plane wave, it is also possible to express τn = ~s · ~rn as a function of the

apparent slowness ~s. Its module is s = 1_c, where c is the module of the apparent velocity of the wavefront. If the stations share the same elevation, sz = 0 and s = _c1_h, with

ch = v/sin(i): i is the incidence angle of the wave from the vertical, while v is the real

velocity of the seismic wave of interest. Thus, chdenotes the apparent horizontal velocity.

The signals at each station of the array represent a sample of the wavefield in space

5

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and time, thus it seems reasonable to ask how efficient in that regard is a given array. To that end, assume a noiseless plane wave with no attenuation, crossing stations dis-tributed over a homogeneous area. In such case, the difference between the waveforms at different stations is only given by the time difference τn = ~s0 · ~rn, where ~s0 is the

correct slowness: this ensures that the resulting beam is the ’best’ one. It is possible to measure the deviation with respect to it caused by the use of any other slowness ~s, by examining the energy of the beam built upon a ’relative’ delay τn= ~rn· ( ~s0− ~s). Using

Parseval’s theorem and the time-shifting properties of the Fourier transform:

E(ω, ~s0− ~s) = 1 2π Z ∞ −∞ | ˆw(ω)|2 1 N N −1 X n=0 eiω ~rn·( ~s0−~s) 2 dω (2.27)

where ˆ denotes the Fourier transform. E can also be expressed as function of the wave-number by defining the transfer function of the array:

C( ~k0− ~k) 2 = 1 N N −1 X n=0 ei ~rn·( ~k0−~k) 2 (2.28)

where ~k = ω~s. This means that when the wavenumber is correctly estimated - that is, ~

∆k ≡ ~k0− ~k = 0 - the energy of the beam is maximized while is attenuated in any other

case. Therefore, an optimal transfer function is one with a narrow, circular central lobe and distant side lobes: these two factors depend, respectively, on the aperture D and the average spacing d between stations. In other words, a ’good’ array is one with little aliasing and high resolution. In terms of wavelength, this translates to 2d < λ < D. If a specific frequency band is chosen, a transfer function can also be mapped against ∆s1 and ∆s2. In figure 2.1an example for two array configurations is given. Their

response function is given in terms of differences both in the k and s domains: in the last case, a half-octave band centred around 1 Hz is used. The first array’s design grants an isotropic and non-aliased response, as opposed to the second one (at least in the depicted difference ranges).

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Figure 2.1: Examples of two arrays for the f-k and their transfer functions.

Now the f-k method of analysis for the estimation of ~k, or ~s, is examined. It basically consists in a beamforming procedure carried out as a grid search on k1 and k2, aimed at

finding and selecting the combination that maximizes the beam power. This is usually accomplished in the spatial and temporal frequency domain. This is better understood by examining the case of a monochromatic plane wave traveling in a one dimensional space such as w(x1, t) = Aei(2π ˜f t+˜k1x1). Its Fourier transform can be computed as:

ˆ w(f, k1) = ∞ Z −∞ ∞ Z −∞

w(x1, t)e−i2πf te−ik1x1dx1dt = Aδ(k1− ˜k1)δ(f − ˜f ) (2.29)

This maps the signal at the n ˜f , ˜k1

o

point in the Fourier space, whose angle gives the apparent velocity of the wave. This can be easily extended to include the dimension k2,

therefore obtaining the BAZ of the incoming wave from the ratio ˜k1/˜k2 and the

mod-ule of ˜k. Of course, a real signal would be very different from the one here examined, exhibiting a much more complex spectrum: however, the premise of finding the point of maximum energy of the beam remains valid. A signal could also be divided into overlapping time slices, so that the f-k technique bears a time-dependent result.

As a final note, by limiting the analysis in a specific frequency range, centred around a frequency f , the analysis can also be carried out in the slowness domain: this yields

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the components of the slowness vector, which in turn give the module of the apparent horizontal velocity and BAZ of the incoming wave. In this case, for a defined frequency band, a choice on the limit and grid-spacing of the slowness domain is necessary [11]. The former is determined by the minimum velocity expected; the latter by the resolu-tion of the array, that is, the size of the central lobe of the slowness-dependent transfer function. This is governed by the ratio between _D1 and the frequency f .

To conclude, the assumptions and requirements for the method are summarized here:

• Stations must be synchronized, so that the differences among waveforms are only due to the features of the wavefields.

• Geological homogeneity of the site and absence of scattering, so that the assump-tion of a plane wave with a definite ~k holds (spatial coherency).

• Due to the Nyquist sampling theorem, d ≤ λ/2, where λ is the minimum interesting wavelength of the signal.

• The largest λ (smallest spatial frequency) is limited by the aperture D of the array: λ < D.

In the following, Obspy [21] routines will be used to actually perform the f-k analysis.

2.3 The Microseism

In this section, a short review of ocean ambient noise is given, since this is the signal that will be analyzed in section 4.3. A method to compute theoretical noise sources is also described, since this will be employed in section4.3.3.

Continuous, small-amplitude background signals are recorded by seismometers all around the planet: such signals are not induced by earthquakes and are therefore termed Am-bient Noise. Global minimum and maximum noise levels have been modelled [22]: these are known as New Low and New High Noise Model (NLNM, NHNM) and are routinely referenced when reporting on the performance of seismic instrumentation [9]. A method to characterize the ambient noise at a site is to perform a Probabilistic Power Spectral Density (PPSD) [23]: this is basically a power spectrum, where a probability is mapped to each frequency bin and amplitude value. The result can then be compared to the global noise model: a practical example is given in figure4.2.

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Such noise has different origins, whether anthropic or natural: however, the most pow-erful contribute is linked to ocean (or sea) waves. This Ocean Ambient Noise is concen-trated in the [1, 20] s period band and can present amplitudes from 1 µm to 10 µm: hence the name Microseism [12]. Given its source, it can exhibit a seasonal variation and is stronger near large water basins: it may even become dominant in seismograms during sea-storms [9].

2.3.1 Mechanisms of Generation

Microseism can be divided in two main categories, Primary and Secondary.

Primary microseism is found at longer periods (> 10 s) and is weaker: it originates from interactions between ocean waves and a shallow, sloping sea-floor. Due to the coupling mechanisms involved, the seismic noise frequency is equal to that of the waves that originate it [12], [24].

Secondary microseism is stronger and concentrated in the [1, 10] s period range. As predicted by theory [25], [11], this is linked to pairs of opposing sea-wave trains. In fact, when these have the same frequency fw and nearly opposite direction (that is, the sum

of wave-numbers is ~kw ' 0), a double frequency pressure field Fp over the sea surface is

generated. This propagates down to the sea-floor with no attenuation and forces seismic perturbations with frequency f = 2fw and ~kw' 0.

At this point, it appears useful to introduce some way to describe the sea-state. This is done through the directional wave spectrum F (fw, θ), that can be factorized as follows:

F (fw, θ) = E(fw)M (fw, θ) (2.30)

E(fw) is the power spectral density of sea surface elevation, it is usually measured by

buoys and has units m2/Hz. M (fw, θ), is the normalized angular distribution of the sea

elevation for each frequency. It is possible to use them [25] to compute the pressure field at f = 2fw and ~k = ~kw' 0: Fp(f, ~k) 2fw,~kw = ρ2_wg22fwE2(fw)I(fw) (2.31)

where ρw is water density and g is the gravitational acceleration. The non-dimensional

factor I(fw) is basically the integral of M (fw, θ) over all directions. The units of Fp are

Pa2 m2 s.

This spectrum, along with other sea-state variables, is available for downloading at ftp://ftp.ifremer.fr/ifremer/ww3/HINDCAST/MED. The database is maintained by

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the IOWAGA6 project, from the Institut Fran¸cais de Recherche pour l’Exploitation de la Mer (IFREMER). In this thesis, the pressure spectrum Fpof the Mediterranean Sea is

used. The relevant filename is p2l: it consists of a four-dimensional array that depends on longitude, latitude, marine frequency fw and time. The times and geographical

co-ordinates arrays have resolutions of three hours and 0.1◦ respectively. The frequency array has 30 elements which span the range [0.05, 0.8] Hz.

2.3.2 Modelling of P Body-Waves Noise Sources

The pressure spectrum Fp can be used to model and compute theoretical microseism

spectra and sources: examples can be found in [26], [25]. In the following, site effects for P waves are computed in order to model P body-waves noise sources over time, as described in [24] and [12] respectively.

First of all, consider a 1D model, consisting of a water layer of depth ∆h and bulk velocity αw, over a crustal half-space of density ρc and velocities αc, βc. As stated in

section2.3.1, a pressure field acting on the sea-surface is responsible for the microseism. A perturbation leaving from a generic point-source towards the sea-floor can be modelled by a P plane wave. This can be represented by its scalar potential:

φ(x1, x3, t) ∼ eiω(px1+ηwx3−t) (2.32)

Here, p denotes the horizontal apparent slowness, while ηw is the apparent vertical

slowness in water and ω = 2πf = 4πfw. If iw is the takeoff angle of the wave, then

ηw = cos(iw)/αw and therefore:

ηw = s 1 α2 w − p2 _(2.33)

Before being transmitted into the crust (with transmission coefficient T (iw)), the wave

gets reflected n times at the water-air and water-crust interfaces, with reflection coef-ficients (−1)n and (R(iw))n respectively. It is worth reminding that T (iw) and R(iw)

depend on additional parameters, that is, the densities and seismic velocities in the crust and water layer. An additional phase Φw is acquired from crossing the water column:

Φw = 2ω

cos(iw)∆h

αw

(2.34)

6

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All the reflected waves can be summed together and multiplied by T (iw) to compute the

potential of the wave transmitted into the crust:

φc(x1, x3, t) = " T (iw) ∞ X n=0 (−1)n(R(iw))neinΦw # φ(x1, x3, t) ≡ C(i_w, ∆h, f )φ(x1, x3, t) (2.35)

Since the module of its argument is less than unity, the series converges to the value:

C(iw, ∆h, f ) =

T (iw)

1 + R(iw)eiΦw

(2.36)

If T (iw) is the P to P transmission coefficient, equation 2.36 represents the site effects

for P body waves7. The transmission and reflection coefficients are obtained by impos-ing the followimpos-ing conditions at the sea-crust interface: continuity of normal stress and normal displacement and null tangential stress. An expression for the coefficients can be found in [24].

By following [12], equation2.36 is extended by including bathymetry and lateral struc-tural variations in the crust: thus, ∆h, ρc, αc and βc become a function of position

in space, namely longitude and latitude. In this thesis, bathymetry is taken from the ifremer database. The geophysical parameters are from the CRUST1.0 model, which is a 3D model of the planet’s crust [27]: more specifically, the uppermost layer of the model is used, following [12]. Thus, C becomes a map, gridded over the geographical coordinates.

Finally, a map of the sources can be obtained at each time and frequency by multiplying the site effects to the pressure spectrum [12], [24]:

S(x₁, x2, f, t) = 2ρc ρw |C| 2 Fp (2.37)

It is worth noting that, in this expression, the dependance from iw was dropped. In

prin-ciple, the relevant angle for the source-receiver couple depends on the position {x1, x2}

of the source itself. A raycasting method may be used [12] to compute this angle. How-ever, for simplicity’s sake, near-vertical incidence is assumed in this thesis across all the geographical grid, that is iw 1.

7_{This can be extended to SV body waves, when a P to SV transmission coefficient is employed instead.}

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2.3.3 Motivations

Studying the microseism may be interesting to clarify its generative mechanisms. For example, the origin of body-waves in ocean seismic noise is still not well established [24]. Moreover, water cannot sustain shear deformations but Love waves have indeed been reported [4] as a consistent portion of the microseism: how and where these are originated is therefore a legitimate question.

However, microseism is also interesting for its geophysical applications. It represents the basis for the technique of ambient noise tomography. In fact, the Green function can be extracted from the noise cross-correlation between two stations, leading to dispersion curves for the measured wavefield. Then, these can be used to infer the structural fea-tures of the Earth between the two stations as a function of depth. The chief assumption of this method is that of an isotropic distribution of the sources. However, the micro-seism is strongly anisotropic, since some regions of the sea-floor may be more prone to exciting noise than others: this can affect the accuracy of the resulting Green function. Therefore, in order to correctly estimate and account for the uncertainty induced by source anisotropy, the direction (or BAZ) of the wavefield must be estimated [12], [11]. In Chapter4.3.2this is achieved by exploiting a 4C analysis technique.

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Chapter 3

Analysis of 4C Station Data

This chapter is dedicated to the analysis of earthquake-induced ground motions, as sensed by a 4C station consisting of a laser gyroscope and a seismometer, installed be-neath mount Gran Sasso (Italy). This analysis is structured in two parts: first, the BAZ estimate of the incoming wavefield; then, once the local BAZ is known, the computation of aT and the phase velocity estimate by means of equation2.5. The aim is to study Love

waves phase velocity as a function of frequency, that is the dispersion analysis described in section2.2.1. This represents a basis for a further study of the local velocity structure at the Gran Sasso, since dispersion curves can be inverted in order to reconstruct the shear velocity as a function of depth: this, however, goes beyond the scope of the present thesis.

The chapter is structured as follows: in the first section, the instruments and their location will be discussed; then, the analyzed data is presented, along with a short de-scription of the preliminary processing steps performed. Two sections are dedicated to the analysis itself, while the results are discussed in the conclusions.

3.1 Experimental Setup

The 4C station used is made up of two colocated instruments. The first one is a tra-ditional broad-band seismometer, a Nanometrics Trillium-240 s, employed to record ground velocities with respect to three axes: East, North and Vertical (or Z). This in-strument belongs to the national seismic network IV, deployed by the Istituto Nazionale di Geofisica e Vulcanologia (INGV): its station-code is GIGS. Additional technical spec-ifications are listed in table3.1.

The second instrument is a large ring laser gyroscope, named GINGERino: it is used to 20

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Analysis of 4C Station Data 21

measure the rotation rate Ωz. The apparatus was installed and is currently maintained

by researchers of the Istituto Nazionale di Fisica Nucleare (INFN). Its operational prin-ciple is the Sagnac effect: in short, a laser gyroscope is essentially a ring optical cavity with two counter-propagating laser beams. When the cavity is rotating, the frequen-cies of the two beams are shifted in opposite directions. Thus, when combined on a photodetector, they produce a figure of interference whose beating frequency fSagnac is

proportional to the rotation rate [1]:

fSagnac= 4

~ A · ~Ω

λgP

(3.1)

where ~Ω is the rotation rate vector, ~A is the oriented area of the optical cavity, P its perimeter and λg is the wavelength of the laser used. In this case, since ~A is vertically

oriented, the instrument is sensitive to Ωz only [1]. It is worth repeating that, in such an

instrument, there are no movable mechanical parts, and thus the measured rotations are not perturbed by translations. Further specifications of the instrument are presented in table 3.1.

GIGS Model Sampling Rate [Hz]

Nanometrics Trillium-240 s 100

Channels Sensitivity [V · s/m]

E-N-Z 1200

GINGERino Optical Medium Side [m]

He-Ne (λg = 680 nm) 3.6

Channels Sensitivity [nrad s−1Hz−12]

Z 0.5

Table 3.1: Specifics of the 4C station employed.

It also bears mentioning that the station was upgraded in mid-2019 with two 24-bit digital recorders Centaur from Nanometrics. These ensure that all four channels are sampled simultaneously, with an accuracy of ≈ 100 µs. This is achieved thanks to a Network Timing Protocol (NTP): the sampling rate used, also reported in table 3.1, is 100 Hz.

The setup is placed within the facilities of the Laboratori Nazionali del Gran Sasso (LNGS) of the INFN, at ≈ 1 km of depth. The underground setting poses a question regarding the validity of equation2.5, since it is obtained for a wave propagating on the surface. Seismic velocities of ≈ 3 km/s can be fairly assumed: for frequencies lower than ≈ 1 Hz, this translates to wavelengths greater than 3 km. Therefore, from now on, the depth of the site will be treated as negligible, at least for lower frequencies.

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In addition, the epicentre is considered to be far enough that a plane wavefront can be assumed.

3.2 Dataset and Preliminary Processing

Event no. Region Magnitude Lat. [◦] Lon. [◦] Depth [km]

1 Northern Albania 6.2 Mw 41.3998 19.5207 22

2 Northern Albania 5.4 ML 41.5355 19.4663 20

3 Bosnia and Herzegovina 5.3 mb 43.1631 17.981 10

4 Crete, Greece 6.0 Mwp 35.6449 23.2515 20

5 Mugello, Italy 4.5 Mw 11.31 44 9

Table 3.2: List of numbered earthquake events analyzed, in chronological order.

Table3.2reports a list of all the earthquake events analyzed in this chapter, with their magnitude, coordinates and depths. For all of them, the same pre-processing is applied. The velocity traces are first pre-filtered, then an instrument response correction is ap-plied. The gyroscope raw data is demodulated by means of its analytical representation in order to obtain the instantaneous frequency: this is then converted to rotation rate by division with the geometrical factor 4A/(λgP ) contained in equation3.1. The data is

presented, for each event in chronological order, in figures 3.1,3.2,3.3,3.4,3.5. To ob-tain these plots, a bandpass filter is first applied, in order to visually enhance the onsets of the seismic phases: P and S arrivals are clearly visible, later to be reached by surface waves. However, P arrivals can be recognized in the rotational trace, too: a mechanism of P-SH conversion at the site must be then involved, since P-polarized phases do not produce vertical rotations [8]. The arrivals labeled in the figures are computed thanks to an Obspy routine, essentially a raytracing algorithm based on the 1D Earth velocity model iasp91. Their function is to constitute a visual aid in the following analysis.

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Figure 3.1: Translational and rotational waveforms for Event n. 1. Traces are filtered in band [0.02, 5.00] Hz for visualization purposes. Theoretical P and S first arrivals are

depicted.

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depicted.

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depicted.

3.3 BAZ Analysis

Here, the BAZ analysis is performed. The procedure is described in more detail for the Albania event of Mw 6.2, however, basically the same workflow is applied to all events of the dataset.

First, the data is band-pass filtered, then the zero-lag correlation matrix C is built, as described in section 2.2.1: the time-step chosen is always ten times longer than the shortest period of the band. After a series of trials, an optimal band for filtering is found: namely, the one that ultimately yields the highest values in C and the least un-certainty on the final BAZ estimate. For the first Albania event, this band lies between 0.04 Hz and 0.25 Hz. These corner frequencies may slightly vary for the other events examined, but are always chosen so that the assumption of near-surface measurements is more likely to hold (see section3.1): they are all reported in table3.3. The C map is shown in figure3.6, bottom panel. An increase in the correlation values can be noticed as soon as the P phase is converted to SH-waves. Since the boundary conditions in this case are different from the ones that give equation 2.5, the high-correlation region corresponds to an angle much lower than the theoretical BAZ. Then, this region moves

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up to angles closer to the theoretical one, but still relatively low. An explanation1 for this will be given in section 3.5. Finally, as the waveform decreases in amplitude, the coherency is lost and the correlation values diminish. Generally, it should be noted that, for the whole duration of the signal, very high correlation coefficients are reached (up to ≈ 0.94).

Figure 3.6: Top panel: aT and Ωz for Event n.1. Bottom panel: C matrix, picked maxima, theoretical and observed BAZ values. The bands used for each panel are

shown.

The next step is to estimate the BAZ. For this purpose, a time window is defined with the aid of the labeled arrivals. The start-time of such a window is set just before the S first arrival, in order to ensure that the surface waves are included for most of their duration: this way, a region is chosen where equation 2.5 holds, at least in principle. In reality, a de-coherency in C can be noted after the S arrival: this may be explained by the presence of other SH polarized waves, for which, however, the usual equation is not valid. Nevertheless, maxima of C are picked in this region: the corresponding BAZ points make up a sample, of which a mean and standard deviation are computed. This represents an estimate of the local BAZ for this event. The value found is θ = 77◦± 23◦, in accordance within slightly more than one standard deviation with the theoretical one θth = 101◦. This estimate is used to rotate the horizontal ground accelerations and

derive aT, according to equation 2.7. This is compared to Ωz in the top panel of figure

3.6.

1

For the time being, suffice it to say that the BAZ at the receiver does not necessarily correspond to the theoretical BAZ, which is the BAZ at the source.

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Figures 3.7, 3.8, 3.9, 3.10 report a similar analysis for the other four events, while table3.3sums up the results found. A general underestimate with respect to theoretical BAZ can be noted for most of the events: an exception is the earthquake in Bosnia-Herzegovina, for which, coincidentally, the BAZ estimate is more precise. Some possible explanations for this will be given in the conclusions.

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shown.

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shown.

Event no. Frequency Band [Hz] Est. BAZ [◦] Th. BAZ [◦]

1 [0.02, 0.25] 77 ± 23 101

2 [0.02, 0.25] 76 ± 17 100

3 [0.02, 0.25] 72 ± 10 76

4 [0.04, 0.25] 100 ± 21 129

5 [0.05, 0.5] 280 ± 24 314

Table 3.3: Summary of the results obtained for the BAZ analysis, along with frequency band used.

3.4 Dispersion Analysis

Once aT is derived, it can be compared to Ωz in order to obtain the phase velocity by

using equation2.5. By splitting the analysis in different period bands, it is possible to study the dispersion of the seismic waves, as explained in section 2.2.1. Each band is centred on periods linearly spaced from 0.2 s to 1 s, then from 2 s to 9 s. The bandwidth is half an octave. As before, a more in-depth comment is given for the Albania event of Mw 6.2, but the same method of analysis is applied to all other events.

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Figure 3.11 shows a comparison between aT and Ωz for different central periods, along

with the result of the phase velocity estimate. Amplitudes are normalized with respect to the maximum value overall. At lower periods, surface waves are dominant, as they can be recognized by their normal dispersive behaviour: shorter period components are slower than their longer period counterpart since they are more sensitive to the shallower layers of the Earth, characterized by lower shear velocities values. At shorter periods (≈ 1 s and lower), body waves (whose arrivals are conveniently labelled) are dominant and, expectedly, they appear to be non-dispersive. These observations are only par-tially reflected by the computed phase velocity values: this could be due to limits in the method of analysis or to un-modelled effects in the propagation. A more in-depth discussion can be found in the conclusions.

Figures 3.12, 3.13, 3.14, 3.15 show the same analysis for the other events. It must be noted that, in some cases, the SNR for Ωz becomes significantly low, so much so as to

invalidate the analysis in some bands: for example, the worst case is Crete 6.0, where noise becomes significant at periods shorter than 0.5 s.

Figure 3.11: Dispersion analysis for Event n.1. The central periods of each band are on the left-hand of the plot. On the right-hand of the plot, c estimates from

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peak-envelope rates are listed for each band.

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peak-envelope rates are listed for each band.

Analysis of Rotational Motions induced by Earthquakes and Ocean Noise obtained by direct Ring laser measurements and array techniques.