Analysis of Earthquake-induced Ground Rotation Signals recorded by GINGERino and G Ring Laser Gyroscopes

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Dipartimento di Scienze della Terra

Corso di Laurea Magistrale in Geofisica di Esplorazione e Applicata

Tesi di Laurea Magistrale

Analysis of Earthquake-induced

Ground Rotation Signals recorded by GINGERino and G

Ring Laser Gyroscopes

Candidato:

Paolo Manganello

Relatore:

Prof. Gilberto Saccorotti

Correlatore:

Dr. Andreino Simonelli

Controrelatore:

Prof. Isidoro Ferrante

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Contents

LIST OF TABLES ... VII

LIST OF FIGURES ... VIII

INTRODUCTION ... 1

Preface ... 1

Purpose and organization of the work ... 3

CHAPTER 1 ROTATIONAL GROUND MOTION: THEORY AND

OBSERVATIONS ... 5

1.1 Rotation induced by seismic waves ... 5

1.2 Historical earthquake rotational effects... 9

1.3 Rotational seismic models ... 12

1.4 Measurements of rotational ground motion ... 16

CHAPTER 2 INSTRUMENTATION ... 19

2.1 Rotation sensors... 19

2.2 Evolution of rotation sensors: from mechanical sensors to optical devices ... 20

2.3 Optical sensors ... 23

2.3.1 Fibre Optic Gyroscopes ... 24

2.3.2 Ring Laser Gyroscopes... 26

2.4 Pioneering studies with Ring Laser Gyroscopes ... 30

2.5 Instrumental apparatus ... 33

2.5.1 GINGERino – Gran Sasso National Laboratories ... 34

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Contents

CHAPTER 3 FUNDAMENTAL METHODS FOR THE JOINT

ANALYSIS OF ROTATIONAL AND TRANSLATIONAL GROUND MOTION

DATA ... 37

3.1 Languages used for data processing ... 37

3.2 Preprocessing and band-pass filtering ... 37

3.3 Rotation of the horizontal components of acceleration ... 39

3.4 Spectral analysis ... 40

3.5 Zero-lag cross-correlation coefficient ... 41

3.6 Back-azimuth estimation ... 42

3.7 Horizontal phase velocity estimation ... 43

3.8 Narrow band-pass filtering and Love wave dispersion curves ... 43

CHAPTER 4 TIME-FREQUENCY ANALYSIS ... 44

4.1 Cohen’s Class ... 44

4.2 Properties of Time-Frequency Distributions... 46

4.3 Uncertainty principle ... 48

4.4 Short-Time Fourier Transform and Spectrogram ... 49

4.5 Wigner-Ville Distribution ... 50

4.6 Choi-Williams Distribution ... 52

4.7 Reduced Interference Distribution – Bessel kernel ... 53

4.8 Application to synthetic signals ... 54

4.8.1 Sine wave ... 54

4.8.2 Quadratic chirp ... 56

CHAPTER 5 DATA ANALYSIS AND RESULTS ... 60

5.1 Analysed earthquakes... 60

5.2 Rotation of the horizontal components of acceleration ... 63

5.3 Power Spectral Density ... 64

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Contents

5.5 Zero-lag correlation coefficient ... 70

5.6 Back-azimuth estimation ... 73

5.7 Horizontal phase velocity estimation ... 75

5.8 Love wave dispersion curves ... 75

5.9 Peak ground rotation and acceleration ... 82

CONCLUSIONS ... 85

APPENDIX A

LINEAR ELASTICITY THEORY AND SEISMIC

WAVES ... 86

A.1 Body waves ... 88

A.1.1 P waves ... 89

A.1.2 S waves ... 90

A.2 Surface waves ... 91

A.2.1 Rayleigh waves ... 92

A.2.2 Love waves ... 93

APPENDIX B

SAGNAC EFFECT ... 95

APPENDIX C

PROGRAMMING LANGUAGES ... 99

C.1 MATLAB®_{... 99}

C.2 ObsPy-pythonTM_{... 100}

C.2.1 PythonTM_{... 100}

C.2.2 Python libraries ... 100

C.2.3 Interactive Python shell ... 101

C.2.4 ObsPy ... 102

BIBLIOGRAPHY ...104

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List of Tables

Table 4.1: Kernel functions for studied Cohen‟s time-frequency distributions………...46

Table 5.1: Parameters of the analysed earthquakes: date, UTC origin time, latitude of epicentre (negative values to the South), longitude of epicentre (negative values to the West), moment magnitude, epicenter location, hypocenter depth (http://cnt.rm.ingv.it/en), peak vertical rotation rate (PRR) and peak horizontal ground acceleration (PGA) (in black peak values of LNGS data, in blue peak values of Wettzell data)………..60

Table 5.2: List of 33 local earthquakes of the October-November 2016 Central Italy seismic sequence recorded by LNGS instrumentation and analysed in this work………...61

Table 5.3: Epicentral distances (in km) and theoretical back-azimuth (in degrees) calculated with the gps2dist_azimuth ObsPy module, and estimated back-azimuth via the method described in 3.7 of the five analysed seismic events recorded by LNGS and Wettzell station…….63

Table 5.4: Peak values of the zero-lag correlation coefficient (ZLCC) and Signal-to-Noise ratio (S/N) for both transverse acceleration and vertical rotation rate calculated for the five analysed events recorded by LNGS and Fundamentalstation Wettzell……….73

Table 5.5: Range of the estimated horizontal phase velocities (in km/s) for the analysed teleseismic events recorded at both LNGS and Wettzell stations………77

Table 5.6: Average and standard deviation of PGA/PRR ratios for the thirty-three local earthquakes recorded by LNGS instrumentation and the five teleseimic events recorded by LNGS and Wettzell stations………...84

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List of Figures

Figure 1.1: Preferred nomenclature and sign conventions for observed translational and rotational motions in seismology and earthquake engineering. X and Y axes point into the page and are in the horizontal plane. Axes point in the positive directions of both the translational vectors ( ) and rotation vectors ( ) (from Evans et al., 2009)………6 Figure 1.2: Cartesian and spherical polar coordinates for analysis of the radial, i.e., the horizontal direction along the source-station path, and transverse components of displacement and rotation induced by a shear dislocation of area A and average slip 〈 〉. The dislocation is in the x1x2 plane, with slip along x1 (from Suryanto, 2006)……….9

Figure 1.3: Schiantarelli‟s drawing of the two rotated obelisks of the monastery of Serra San Bruno induced by the 1783 Calabrian earthquake (from Kozák, 2009)………...10 Figure 1.4: Examples of earthquake rotated objects. Left: obelisk erected in 1850 at Chatak, India, which broke and rotated counter-clockwise during the 1897 Assam earthquake (from Kozák, 2009). Right: dislocated and counter-clockwise rotated tombstones in the Kushiro Cemetery after the 2003-09-26 Tokachi-Oki earthquake (from Hinzen, 2012)………..11 Figure 1.5: Gray‟s model showing translational motion impinging (denoted by the inward

pointing arrows) on a column, in plane view. G is the center of gravity. “If the shock has some intermediate direction, such as e G, and its effect would be to cause the

body to bear heavily on B and at the same time to rotate around B as an axis in the direction of the hands of a watch, i.e. from right to left”. Gray proposed that a shock passing through any of the shaded octants in the figure would cause rotation from right to left, whereas if the shock passed through any of the unshaded octants, rotation from left to right would occur (using this model Meldola & White were not able to constrain the propagationdirection) (Sargeant&Musson,2009)………..14 Figure 1.6: Fracture planes and rotation effects due to friction processes or dislocations motions. On the left, the symmetric case; on the right the fracture along a main fault. The arrows indicate the shear stresses acting along the fracture planes (Teisseyre et al., 2003)………..……..15 Figure 2.1: Upper left: detail of the prototype Cecchi electrical seismograph with sliding smoked paper (after restoration), showing the recording system (on the left) and the apparatus for the surveying of rotational movements (from Ferrari, 2006). Upper right: rotation-seismograph system consisting of two oppositely oriented seismographs, having pendulums suspended on a common axis; this instrument can record rotation, twist and stretching effects (from Teisseyre et

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List of Figures

al., 2003). Bottom left: the S-5-SR seismometer recording rotational components of ground

motions around the vertical axis (behind) and around the horizontal axis (in front); both sensors without covers (from Kaláb et al., 2013). Bottom right: a prototype of the seismic rotational sensor system, Rotaphone, containing eight horizontal geophones (LF-24 made by Sensor Nederland b.v.) in four opposing pairs, deployed at the station Nový Kostel in West Bohemia (Czech Republic) and intended to measure the vertical rotation rate and the two horizontal components of the ground translational velocity (from Brokešová et al., 2012)………….…………22 Figure 2.2: Configuration of open-loop fibre optic gyroscopes (on the left), and closed-loop fibre optic gyroscopes (on the right) (from Fidanboylu & Efendioğlu, 2009)………..…25 Figure 2.3: Ring Laser Gyroscope diagram (from Ayswarya et al., 2015): two counter-propagating single-mode laser beams interfere to generate a beating if the gyroscope rotates with respect to the surface normal. The beating frequency is directly proportional to rotational rate………..27 Figure 2.4: Large ring-laser gyroscopes at Cashmere, Christchurch, New Zealand: C-I (upper left) and C-II (upper right) (from Stedman, 1997). Below left: GEOsensor ring laser during sensor integration (Schreiber, Stedman et al., 2006). Below right: GP2 apparatus (from Simonelli, 2014)……….31 Figure 2.5: Location of the Gran Sasso National Laboratories (Central Italy) and Geodetic Fundamentalstation Wettzell (SE-Germany) (map created with Python toolkit Basemap using a blue marble image from NASA on a Mercator projection)………...………..34 Figure 2.6: On the left: map of the Gran Sasso National Laboratories (Italy) and position of GINGERino apparatus; on the right: the GINGERino ring laser gyroscope (from Simonelli et al., 2016)……….………35 Figure 2.7: On the left: the Nanometrics Trillium 240 s and the Lippmann 2-K tiltmeter (smaller metal case on top); the red arrow shows the North direction. On the right: the Guralp 3T360s seismometer (from Simonelli et al., 2016)………..35 Figure 2.8: The G ring laser gyroscope at the Geodetic Fundamentalstation Wettzell, Germany. Left: cross-sectional view of the instrument site (from Lee et al., 2012); right: photo of G ring laser gyroscope (from Igel et al., 2007)……….……….36 Figure 3.1: Coordinate system for the rotation of the horizontal components (N-S and E-W) of translational ground motions. The back-azimuth θ (BAZ) is the angle between the incoming seismic wavefront and the geographical North measured generally clockwise at the receiver station. Azimuth angle is equal to BAZ+180°………39

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List of Figures

Figure 4.1: Comparison between spectrograms of the considered sine wave calculated using two different Hamming window lengths, 0.56 s and 5.92 s. Also reported is the sine wave in both the time domain (top) and frequency domain (left)……….55 Figure 4.2: Time-Frequency Distributions of the considered sine wave: (a) Spectrogram, (b) Wigner-Ville Distribution, (c) Choi-Williams Distribution, (d) Reduced Interference Distribution (Bessel)……….………56 Figure 4.3: Time-Frequency Distributions of the considered quadratic chirp: (a) Spectrogram, (b) Wigner-Ville Distribution, (c) Choi-Williams Distribution, (d) Reduced Interference Distribution (Bessel). Also reported above is the quadratic chirp in time domain……….58 Figure 5.1: Location of LNGS (Italy), Fundamentalstation Wettzell (Germany) and epicenters of the five studied earthquakes, and wavefront path on an orthographic projection of the Earth (plot created with Python toolkit Basemap). The stations are marked by red triangles; the epicenters are marked by red stars………..………62 Figure 5.2: Map showing the epicentral locations (red stars) of the analysed 33 local events of the October-November 2016 Central Italy seismic sequence. The red triangle marks the location of the Gran Sasso National Laboratories. This map has been created with Python toolkit Basemap using an image from the ArcGIS Server REST API (ESRI Imagery World 2D) on a Gauss-Boaga projection……….……….62 Figure 5.3: Power Spectral Density estimates for LNGS data of the Mw 6.5 2015-11-17 Greece

earthquake vertical rotation rate (top) and transverse acceleration (bottom) times series of a 2h long noise window preceding the earthquake, a 90 s long window including the S-wave and coda, and a 600 s long window starting at the Love wave arrival. The power spectra estimates are computed with the Welch method by the averaging and squaring of the spectral estimates obtained over a 45 s long sliding window with 50% overlap. Before transformation, individual signal slices were demeaned, detrended and tapered by a Hamming window………….…………..65 Figure 5.4: Power Spectral Density estimates for the Mw 6.7 2015-11-13 Ryukyu Islands

earthquake vertical rotation rate (top) and transverse acceleration (bottom) times series of a 5h long noise window preceding the earthquake, a 400 s long window including the S-wave and coda, and a 2000 s long window starting at the Love wave arrival. The power spectra estimates are computed with the Welch method by the averaging and squaring of the spectral estimates obtained over a 150 s long sliding window with 50% overlap. Before transformation, individual signal slices were demeaned, detrended and tapered by a Hamming window………66 Figure 5.5: Spectrograms of transverse acceleration (top) and vertical rotation rate (bottom) of the 2015-11-17 Greece earthquake LNGS data, obtained using a 60 s sliding

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List of Figures

window with 90% overlap. Also reported are the associated seismic traces that have been band-pass filtered between 0.01 and 2 Hz……….68 Figure 5.6: Spectrograms of transverse acceleration (top) and vertical rotation rate (bottom) of the 2015-11-13 Ryukyu Islands earthquake Wettzell data, obtained using a 90 s sliding window with 90% overlap. Also reported are the associated seismic traces that have been band-pass filtered between 0.003 and 1 Hz………69 Figure 5.7: Choi-Williams Distribution for vertical rotation rate of the 2015-11-17 Greece earthquake (LNGS data). Frequencies above 1 Hz are not shown. Also reported is the associated seismic trace that has been band-pass filtered between 0.01 and 2 Hz…………70 Figure 5.8: Choi-Williams Distribution for vertical rotation rate of the 2015-11-13 Ryukyu Island earthquake (Wettzell data). Frequencies above 0.5 Hz are not shown. Also reported is the associated seismic trace that has been band-pass filtered between 0.003 and 1 Hz…..70 Figure 5.9: Vertical rotation rate (red, top) and transverse acceleration (black, middle) of the 2015-11-13 Ryukyu Islands (Japan) earthquake recorded by Gran Sasso National Laboratories (after applying routine pre-processing and band-pass filtering); also marked are the P-wave onset (P), the S-wave onset (S) and the surface waves onset (SW). The zero-lag correlation coefficient between the two traces (dashed blue line, bottom) is calculated using a 120 s sliding time window with 50% overlap. Time zero is at 21:00:00 UTC………..………..71 Figure 5.10: Vertical rotation rate (red, top) and transverse acceleration (black, middle) of the 2015-11-13 Ryukyu Islands (Japan) earthquake recorded by Fundamentalstation Wettzell (after applying routine pre-processing and band-pass filtering); also marked are the P-wave onset (P), the S-wave onset (S) and the surface waves onset (SW). The zero-lag correlation coefficient between the two traces (dashed blue line, bottom) is calculated using a 120 s sliding time window with 50% overlap. Time zero is at 21:00:00 UTC……….72 Figure 5.11: Top: Back-azimuth estimation for LNGS data of the 2016-06-26 Tajikistan earthquake using a 120 s window with 50% overlap. The dashed black line is the theoretical back-azimuth, the dashed red line is the estimated back-azimuth. Middle: Superposition of the transverse acceleration trace obtained using the theoretical back-azimuth (black, left axis) and the vertical rotation rate trace (red, right axis). Time zero is at 11:00:00 UTC. Bottom: Comparison of the ZLCCs calculated using the theoretical back-azimuth (black line) and the estimated back-azimuth (red line)………..……….74

Figure 5.12: Top: Back-azimuth estimation for Wettzell data of the 2016-06-26 Tajikistan earthquake using a 120 s sliding window with 50% overlap. The dashed black line is the

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List of Figures

theoretical back-azimuth, the dashed red line is the estimated back-azimuth. Middle: Superposition of the transverse acceleration trace obtained using the theoretical back-azimuth (black, left axis) and the vertical rotation rate trace (red, right axis). Time zero is at 11:00:00 UTC. Bottom: Comparison of the ZLCCs calculated using the theoretical back-azimuth (black line) and the estimated back-azimuth (red line)……….75 Figure 5.13: Horizontal phase velocity estimation for LNGS data of the 2015-11-13 Ryukyu Islands earthquake. Top: Superposition of transverse acceleration (black, left axis) and vertical rotation rate (red, right axis). Time zero is at 21:00:00 UTC. Middle: Zero lag correlation coefficient of the two seismic traces in a 120 s sliding time window (with 50% overlap). Bottom: Best fitting horizontal phase velocities as a function of time in the sliding window. The colour of the symbols indicates the value of the correlation coefficient in the sliding time window, according to the colour scale on the right……….76 Figure 5.14: Horizontal phase velocity estimation for Wettzell data of the 2015-11-13 Ryukyu Islands earthquake. Top: Superposition of transverse acceleration (black, left axis) and vertical rotation rate (red, right axis). Time zero is at 21:00:00 UTC. Middle: Zero-lag correlation coefficient of the two seismic traces in a 120 s sliding time window (with 50% overlap). Bottom: Best fitting horizontal phase velocities as a function of time in the sliding window. The colour of the symbols indicates the value of the correlation coefficient in the sliding time window, according to the colour scale on the right……….77 Figure 5.15: Superposition of transverse acceleration (black) and vertical rotation rate (red) narrow band-filtered signals of the 2015-11-13 Ryukyu Islands earthquake recorded by LNGS, and determination of Love wave phase velocities for every dominant period. The transverse acceleration is scaled by twice the appropriate phase velocity (reported at the top of each trace pair) to fit the vertical rotation rate. The lowermost traces are the broadband seismograms. Signals start 900 s before the Love wave onset, and the time scale is referred to the time zero 21:00:00 UTC………78 Figure 5.16: Superposition of transverse acceleration (black) and vertical rotation rate (red) narrow band-filtered signals of the 2015-11-13 Ryukyu Islands earthquake recorded by Wettzell Fundamentalstation, and determination of Love wave phase velocities for every dominant period. The transverse acceleration is scaled by twice the appropriate phase velocity (reported at the top of each trace pair) to fit the vertical rotation rate. The lowermost traces are the broadband seismograms. Signals start 800 s before the Love wave onset, and the time scale is referred to the time zero 21:00:00 UTC………..79

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List of Figures

Figure 5.17: Love wave dispersion curve (black line) for the 2015-11-13 Ryukyu Islands earthquake obtained from LNGS data. The 95% confidence bounds are also reported (green and red dashed lines)……….80 Figure 5.18: Love wave dispersion curve (black line) for the 2015-11-13 Ryukyu Islands earthquake obtained from Wettzell data. The 95% confidence bounds are also reported (green and red dashed lines)………80 Figure 5.19: Estimated Love wave dispersion curves from LNGS data of the 5 analysed teleseismic events. The dashed black line is the dispersion curve from Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981)………81 Figure 5.20: Estimated Love wave dispersion curves from Wettzell data of the 5 analysed teleseismic events. The dashed black line is the dispersion curve from Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981)………81 Figure 5.21: Average and standard deviation (68% confidence interval) of the estimated Love wave dispersion curves from LNGS data (blue) and Wettzell data (red) for the five analysed teleseismic events. The dashed black line is the dispersion curve from Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981)………82 Figure 5.22: Vertical peak ground rotation rate (PRR) versus horizontal peak ground acceleration (PGA) for the thirty-three local earthquakes of Central Italy seismic sequence and the five teleseismic events analysed in this work, recorded by LNGS instrumentation. Magnitudes (Mw)

are in color scale according to the scale on the right; the size of the symbols is proportional to the epicentral distance. The black dash-dot line is the least-squares fit to the local earthquakes data……….………..83 Figure A.1: Illustration of the dynamics of an earthquake. The body waves propagate from the hypocenter, the point in the Earth where the rupture an earthquake originates, and the epicenter, which is the point on the Earth‟s surface vertically above the hypocenter (from: http://data.allenai.org/tqa/inside_earth_L_0075/)...88 Figure A.2: Sense of particle motions relative to the direction of propagation for P-wave and

S-wave disturbances (from Encyclopædia Britannica, Inc.:

https://www.britannica.com/science/seismograph)...89 Figure A.3: Graph of the average seismic waves velocity and density profiles through the Earth, according to the Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981). The velocities of compressional (VP) and shear (VS) waves are given on the

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List of Figures

left axis, density on the right axis, and pressure (in GPa) as a function of depth (in km) on the top scale (http://eesc.columbia.edu/courses/v1011/topic3/PartThree.html)...91 Figure A.4: Sense of particle motions relative to the direction of propagation for Love and Rayleigh waves disturbances (from Encyclopædia Britannica, Inc.: https://www.britannica.com/science/seismograph)...94 Figure B.1: The Sagnac effect for two counter-propagating light beams in the same rotating cavity (circular for simplicity): (a) at rest in an inertial frame of reference; (b) rotating with respect to an inertial frame of reference. Notation: R is the radius of loop, Lcw

is the distance in clockwise direction, Lccw is the distance in counter-clockwise direction, Ω

is the angular velocity of interferometer rotation (from Jaroszewicz et al., 2006)…………..96

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Introduction

Preface

Modern seismology is still based mainly on the observation, processing and inversion of the three components of translational ground motions (displacement, velocity or acceleration), as recorded by standard inertial instruments. Ground deformation, which in classical continuum mechanics is described by the six components of the strain tensor, is the other type of measurement that is adopted to monitor regional and global seismic wave fields. Indeed, as it has been noted for decades by theoretical seismologists (e.g., Aki & Richards, 2002) and earthquake engineers, in addition to translations and strains, three component of rotations (a vectorial quantity) should also be measured in order to fully characterize ground motion. This allows the reconstruction of the complete motion of a measurement point in a solid body, helping to further constrain earthquake source processes, structural properties, and provide additional information which is relevant to hazard assessment (Igel et al., 2005, 2007; Lee et al., 2007; Lee et al., 2009; Schreiber et al., 2009). This type of measurement has not been carried out due to a widespread belief that rotational motions are insignificant (e.g., Richter (1958)) and, consequently, difficult to measure, especially owing to the lack of high-sensitivity devices. In the absence of direct measurements, ground rotations have been derived indirectly from the spatial gradient of translational motions observed by dense accelerometer or seismometer arrays (e.g., Huang (2003); Spudich & Fletcher (2008)). In such „geodetic‟ determinations of ground rotations, the distance between the individual sensors was usually several tens to several hundreds of meters, which causes specific problems including different site conditions beneath the individual seismographs in the array and validity of the estimates for seismic waves whose wavelength is significantly larger than the distances between the recording sensors. In addition, the accuracy of these observations could never be tested in the absence of direct measurements (Suryanto et al., 2006). The rotational component of ground motion has been also estimated theoretically using kinematic source models and linear elastodynamic theory of wave propagation in elastic solids (e.g., Bouchon & Aki, 1982). It is expected that such estimates are approximately correct in the far-field, but in the near-field, where the soil may experience nonlinear response, the rotational components of strong motion are expected to be larger than the linear estimates, and the need to include rotation estimates in the analysis of motion is even more evident (Lee et al., 2007, Lee, Çelebi et al., 2009).

In recent years the direct measurement of rotational components of earthquake-induced ground motion has become a reality due to high-resolution ring laser gyroscopes. Since these instruments are not based on any moving mechanical component, they are entirely insensitive to

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Introduction

translational motion and are able to measure the rotation rate with high linearity and accuracy over a wide frequency band and epicentral distance range (Todorovska et al., 2008; Schreiber et

al., 2009; Velikoseltsev et al., 2012). Three such devices, mounted in orthogonal orientations,

may eventually provide the quantitative detection of rotations associated with S-, Love and Rayleigh waves. These should evolve into several six-component observatories equipped with both RLG and broadband seismometers (Schreiber, Igel et al., 2006).

During the last decade, several earthquakes were recorded by the large ring laser gyroscopes located in New Zealand, Germany and USA, and the subsequent data analysis demonstrated reliability and consistency of the results with respect to theoretical models from classical elasticity theory. In this regard, the pioneering study of Igel et al. (2005), using co-located observations of three components of ground translation from a broadband velocity seismometer and the vertical rotation rate from the G ring laser gyroscope, suggested that the transverse acceleration and vertical rotation rate should be in phase with relative amplitudes scaled by two times the horizontal phase velocity.

So far, the measurement of rotational motion of the seismic wavefield at teleseismic distances have undergone significant advances with the advent of ring-laser instruments and, show a good agreement with classical elasticity theory. However, the evaluation of ground rotations in the near field (i.e., within about 4 wavelengths from the hypocenter) of strong earthquake (magnitude > 6.5) sources, where the rotations amplitude can be one to two orders of magnitude greater than that expected from linear elasticity theory, have been scarcely measured, with the exception of some works (e.g., Takeo (1998, 2009) and Liu et al. (2009) for earthquakes and Nigbor (1994) and Lin et al. (2009) for chemical explosions). The recording of rotational ground motion in the near field would require extensive seismic instrumentation along some well-chosen active faults as well as a certain degree of luck (e.g., Lee, Huang et al., 2009). It would, nevertheless, represent an important contribution to understanding more about source mechanism and, in particular, in its assessment within the field of seismic engineering regarding its potential in causing earthquake rotated objects (EROs) (Lee, Çelebi et al., 2009; Lin et al., 2011; Hinzen, 2012).

Nowadays, advances in sensor technology have made it possible to execute a six component polarization analysis in a single station comprised of three components of rotational motion and three components of translational motion. This provides the opportunity to unambiguously identify the wave type, propagation direction and local P and S wave velocities at the receiver location, where information is extractable by conventional three component translational processing using only large and dense receiver array data (Sollberger et al., 2018).

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Introduction

Purpose and organization of the work

The aim of this work is to compare earthquake-induced ground rotations observed at the GINGERino and G ring-laser gyroscopes, located at the Gran Sasso National Laboratories and Geodetic Fundamentalstation Wettzell, respectively, with the co-located translational measurements from standard seismometer. By computing the Power Spectral Density (PSD) of vertical rotation rate and transverse acceleration, first, I identify the signal to noise ratio (SNR) as a function of frequency, and then, by applying Time-Frequency Analysis, I individuate the most energetic frequency bands as a function of time for both instruments in relation to five teleseismic events. I then correlate the rotation rate and acceleration in order to quantify similarities between the two signals. The first aim is to obtain the back-azimuth angle estimation (direction of the incoming wavefield) by searching for the correct angle which gives the maximum value of the correlation coefficient between transverse acceleration and vertical rotation rate in a sliding time window. The second aim is to obtain the horizontal phase velocities estimation from the co-located measurements of rotation and translation, and the subsequent derivation of Love wave dispersion curves. This is attained by computing the zero-lag correlation coefficient (ZLCC) between translational and rotational traces, and estimating the phase velocity from a linear regression of the amplitudes of the two signals for those time intervals over which the ZLCC is above a given threshold (Igel et al., 2005). By iterating the procedure over narrow band-pass filtered traces, it is possible to obtain a dispersion curve for the selected wave packet (Igel et al., 2007). The estimated Love wave dispersion curves are compared against the theoretical dispersion curve derived from the Preliminary Reference Earth Model (PREM) of Dziewonski and Anderson (1981).

This thesis contains five chapters. In Chapter 1 I briefly describe the basic theory of rotational seismology, presenting the basic equations of linear elastic waves, and the relationship between rotation and translation in classical elasticity theory. I also report observations of historical earthquake-induced rotational effects and rotational theory models developed from the mid nineteenth century.

In Chapter 2 I present a brief overview of rotation sensors, from the first mechanical sensors to high sensitivity ring-laser technology, also describing the instrumental apparatus of Gran Sasso National Laboratories and Geodetic Fundamentalstation Wettzell, whose data are the object of the subsequent analysis.

In Chapter 3 I describe the fundamental methods adopted in order to carry out the joint analysis of rotational and translational ground motions induced by five teleseismic events recorded by the instrumentation described in Chapter 2.

In Chapter 4, I describe methods for the time-frequency analysis of non-stationary signals, with the main purpose of finding valid alternatives to the classical approach of the Short-Time Fourier Transform. I first introduce definitions and properties of some important time-frequency

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Introduction

distributions, specifically the Wigner-Ville, Choi-Williams and Reduced Interference distributions. I then apply the aforementioned time-frequency distributions to synthetic signals in order to find the best distribution for characterizing the rotational signals in both the time and frequency domains.

In Chapter 5 I report the results of the work and investigate the general relationships between ground rotation and translation by comparing peak values of vertical rotation rate (PRR) with the peak ground acceleration (PGA), an intensity measure commonly used in earthquake engineering, for both teleseismic events and local earthquakes.

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Chapter 1 Rotational Ground Motion: Theory and Observations

Rotational seismology is an evolving field of seismology for studying all aspects of rotational ground motions induced by earthquakes, ambient vibrations and explosions. This subject is of interest to several disciplines, such as earthquake engineering, geodesy and Earth-based detection of Einstein‟s gravitational waves (Lee, Çelebi et al., 2009; Lee et al., 2012).

In this chapter I outline the fundamental theory of rotational seismology, focusing on some of the most important developed rotational seismic models, and include some important studies on the rotational effects of historical earthquakes and measurements of ground motion rotation.

1.1 Rotation induced by seismic waves

The classical textbooks on seismology refuted any possibility of rotational motions, and the rotational effects of the earthquakes were explained by an interaction of seismic waves with a compound structure of the objects they penetrate; nevertheless it was proved theoretically that rotation motions, especially in form of seismic rotational waves (SRWs), could propagate through grained rocks. Later on, this possibility was extended to rocks with microstructure or defects (e.g., dislocations, disclinations, vacancies, thermal or piezoelectric nuclei) or even without any internal structure, due to the asymmetric stresses in the medium (in this case one needs to consider the asymmetric continuum theory) (Teisseyre et al., 2003; Teisseyre, 2012). The effects related to deviations from ideal elasticity depend on the real material properties and the thermodynamical state of a body; for rigid brittle rocks with properties not deviating far from those of ideal elasticity such deviations produced by the self/internal distortion field can be very small (Teisseyre et al., 2003).

Many assumptions have been made on what can cause rotations, for example, by Michael (1987) and Zhao et al. (1997), who suggested the possibility of inelastic processes in the rupture zone; Hauksson (1994), who explained his results in terms of stress refraction on a weak fault zone; and Smith and Dieterich (2007), who found that a heterogeneous 3-D fractal distribution of crustal stresses creates an apparent stress rotation after the occurrence of a large earthquake (Pujol, 2009).

Considering a linear elastic medium, if is the displacement at position (which describes the translational motion of the ground caused by passage of surface wave), the displacement at an arbitrarily close position is given by:

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Chapter 1 – Rotational Ground Motion: Theory and Observations

δ = ( ) ( )

where is the i-th component of the displacement vector u, is the 3×3 symmetric strain

tensor and is the anti-symmetric rotational tensor (pseudovector which does not enter

Hooke‟s law and represents the angle of the rigid rotation generated by the disturbance) (Mulargia, 2000; Legrand, 2003; Igel et al., 2005; Cochard et al., 2006; Pujol, 2009). Explicitly the relation between infinitesimal rotations and translational motions is obtained through the application of curl differential operator:

( ) ( ₎

From (1.2) it follows that the rotation rate is equal to the curl of the particle velocity :

Assuming, as recommended in Evans et al. (2009), a right-handed Cartesian coordinate system with axes in East, North and upward directions, it is possible to ascertain that the X-axis is horizontal and positive to the East, the Y-axis is horizontal and positive to the North, and the Z-axis is vertical and positive upward; in this coordinate system a rotation vector is positive in the direction of the right-hand thumb when the particle motions are in the direction that the wrapped right-hand fingers point (Fig. 1.1).

Figure 1.1: Preferred nomenclature and sign conventions for observed translational and rotational

motions in seismology and earthquake engineering. X and Y axes point into the page and are in the horizontal plane. Axes point in the positive directions of both the translational vectors ( ) and rotation vectors ( ) (from Evans et al., 2009).

𝑢𝑋 (often East or short axis of structure)

𝑢𝑌 (often North or long axis of structure) 𝑢𝑍 (always Up) 𝜃𝑌 𝜃_𝑋 𝜃_𝑍

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Earthquake-induced rotational motions can be divided into two components: tilt (also called rocking by many authors), which is the rotation around one of the two horizontal axis (x- or y-axis); and torsion, which indicates the rotation about the vertical axis (z-axis) (Smerzini, 2008).

In elastic isotropic layered half-space, P, SV and Rayleigh waves will produce only rotational

excitations around horizontal axis (rocking), while SH and Love waves will produce only rotational

excitations about vertical axis (torsion). Rotation angles are generally given in units of radians (rad) (Lee, 2009).

In the simplest case of a plane seismic wave (approximation valid for a teleseism), with, for example, polarization along the y-axis and propagation along x-axis (since the wave is transversely polarized, such as the Love wave), the displacement vector is expressed through the equation:

( ( ) )

where is the horizontal phase velocity (Boffi & Castellani, 1988). The vector of rotation is thus given as:

(

̇ ( ))

with the corresponding z-component of rotation rate given by (Schreiber, Igel et al., 2006):

̈ ( )

This implies, under the given assumptions, that at any time vertical rotation rate and transverse acceleration are in phase and the amplitude are related by:

̈

which provides a direct estimation of the phase velocity (Pancha et al., 2000; Schreiber, Igel et al., 2006). The Eq. (1.7) shows that phase velocity scaling implies that ground rotations are two to three orders of magnitude smaller than the associated translational motions, and consequently the recording of ground rotations needs high sensitivity (Simonelli et al., 2016). Additionally it is interesting noting that when the wave velocity becomes smaller, for example in soft or unconsolidated sedimentary and/or fluid-infiltrated porous media, rotations become comparatively larger (Cochard et al., 2006).

Although this proportionality was well-known, it has only been shown experimentally in the last decade by Igel et al. (2005) for the Tokachi-Oki earthquake (2003-09-25, Mw = 8.1),

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At the free surface, the boundary condition implies that the vertical component of the stress tensor vanishes ), and thus directly applying Hooke‟s law for a homogeneous isotropic medium one find the following conditions for stress components , and at (Cochard et al., 2006; Sollberger et al., 2018):

( ) (1.8)

Using these equations it is possible to express the vertical derivatives in terms of horizontal derivatives in the following way:

( )

Applying the Eq. (1.9) to Eq. (1.2) one obtains the expression of the rotation vector at the free surface:

( )

which show that at the Earth surface the horizontal components of rotation correspond to tilt. Consequently P waves generate horizontal rotation at the surface whereas they are irrotational in the bulk. There is an additional contribution to horizontal rotation due to P to SV converted waves at the surface (Cochard et al., 2006).

Aki and Richards (2002) gave expressions for the displacement field generated by a double-couple point source (i.e. point source shear dislocation) in an unbounded, homogeneous, isotropic and elastic medium:

∫ ( ) ( )

̇ ( ) ̇ ( ) where is the density of the medium, is the P-wave velocity and is the S-wave velocity.

The functions A in each term of Eq. (1.11) represent the radiation pattern given by: ̂ ̂ ̂

_{̂ ̂ ̂}

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_̂

_{̂ ̂}

where ̂, ̂,and ̂ are unit direction vectors in a spherical polar coordinate system for the source receiver geometry, the superscripts N, I and F are related to near field, intermediate, and far field terms respectively, while P and S waves denote P and S waves respectively. The strongest P-wave motion is expected to be at 45° angles to the fault plane x1x2, while the S-wave radiation pattern

is rotated by 45°with respect to P-wave pattern (Fig. 1.2).

Applying the curl operator to the Eq. (1.11) one obtains that rotations are given by:

* ( )

( ) ( )+ where is the radiation pattern of the three components of rotation (with the radial component equal to zero), defined as:

̂ ̂

and 〈 〉 is the time dependent seismic moment, in which is the shear modulus, 〈 〉 is the average slip, and A is the fault surface area. The shear dislocation is assumed located in the center of a Cartesian coordinate system x1x2x3 (Fig. 1.2), where the

fault lies in the x1x2 plane, and the slip direction is along x1 (Cochard et al., 2006;

Madariaga, 2007).

Figure 1.2: Cartesian and spherical polar coordinates for

analysis of the radial, i.e., the horizontal direction along the source-station path, and transverse components of displacement and rotation induced by a shear dislocation of area A and average slip 〈 〉. The dislocation is in the x1x2 plane, with slip along x1 (from Suryanto, 2006).

As intuitively expected rotations are zero at the arrival of P-wavefront, starting only at the S-wave arrival, even in the displacement near-field region.

1.2 Historical earthquake rotational effects

The birth of seismology was closely linked with the occurrence of two strong earthquakes which occurred in Lisbon (Portugal) on 1 November 1755 and in Calabria (Italy) on 5 February 1783. These earthquakes also increased interest in earthquake rotational effects, in fact before

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their occurrence very little was written on the nature of the observational phenomenon of rotational seismological effects (Kozák, 2009).

In particular, after the Calabrian seismic event (intensity XI-XII of

Mercalli-Cancani-Sieberg Macroseismic Scale (MCS)), Pompeo Schiantarelli, an Italian architect, contributed to the recording of the earthquake‟s effects through detailed pictorial records, among which a drawing of the distortion of two decorative obelisks in the monastery of Serra San Bruno (also described by Vivenzio (1783,1788)), which had a mutual rotation or twisting of the four vertically arranged stony blocks composing them around a vertical axis caused by seismic movements (Fig. 1.3). Schiantarelli‟s drawing of the distortion of San Bruno obelisks was the first illustration of the rotational effects of an earthquake and it soon became famous as a symbol of this type of seismic effect (Ferrari, 2006; Kozák, 2009).

Figure 1.3: Schiantarelli‟s

drawing of the two rotated obelisks of the monastery of Serra San Bruno induced by the 1783 Calabrian earthquake (from Kozák, 2009).

Following this, several authors described earthquake-induced rotational effects during the whole nineteenth century and the beginning of the twentieth century within Europe: 1805 Baranello (Italy) earthquake by Poli (1806); 1818 Catanese (Italy) earthquake by Ferrara (1818) and Longo (1818); 1839 Comrie (Scotland) earthquake by Milne (1842); 1857 Basilicata (Italy) earthquake by Mallet (1862); 1857 Gera (Germany) earthquake by Seebach (1873); 1873 Belluno (Italy) earthquake by Sieberg (1904); 1895 Ljubljana (Slovenia) earthquake by Suess (1896); 1914 Linera (Italy) by Sabatini (1914); Platania (1915). Outside Europe: the 1894 Shönai (Japan) earthquake was described by Omori (1894 or 1895) and Sieberg (1904); 1897 Assam (India) earthquake by Oldham (1899) (Fig. 1.4); and the 1906 San Francisco (USA) earthquake by Jeništa (1906-1907)). These observed rotational effects are especially evident on vertically oriented objects like statues, monuments, tombstones, chimneys, columns and obelisks, which were rotated on their supports (Kozák, 2006; Kozák, 2009; Hinzen, 2012).

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It has been conjectured that some of these early observations might have been the effects of ground rocking caused by soil-structure interaction and not the effects of true rotational motion (Lee et al., 2009; Sargeant & Musson, 2009). Some authors suggest that, although translational motion would be able to produce these effects, the fact that the majority of chimneys were twisted in the same direction is evidence for translational movement having played a subordinate role only, proposing rotational waves or at least vertical motions as cause of such rotations (Lee

et al., 2012).

Earthquake rotated objects were associated by Sieberg (1923) in his revision of the Mercalli-Cancani intensity scale with intensity grade VII destructive, as well as in the 1956 version of modified Mercalli scale (MM), while at the end of twentieth century rotations were graded with grade VIII (heavily damaging) in the European Macroseismic Scale (EMS). The advent of more sophisticated seismometers made macroseismic observations no longer necessary to determine the coordinates of seismic events, and as a consequence, less attention was paid to rotated objects during earthquakes. In this regard Gutenberg (1927) and Richter (1958) in their textbooks supported the hypothesis that rotation is not particularly significant in earthquake seismology. Nevertheless, some surveys on earthquake rotated objects have continued to be made during the instrumental period of seismology, such as Hodgson (1950) on the 1925 St. Lawrence, Québec (Canada) earthquake; Ulrich (1936) on the 1935 Helena, Montana (USA) earthquake; Rouse and Priddy (1938) on the 1937 Western Ohio (USA) earthquakes; Lander (1968) on the 1968 North California (USA) earthquake; Gordon et al. (1970) on the 1968 South-Central Illinois (USA) earthquake; Keyser (1974) on the 1969 Tulbagh (South Africa) earthquake; Yamaguchi and Odaka (1974) on the 1974 Izu-Hanto-oki (Japan) earthquake; Mauk et al. (1982) on the 1980 Sharpburg, Kentucky (USA) earthquake; Yegian et al. (1994) on the 1988 Spitak (Armenia) earthquake; Tertulliani and Maramai (1998) on the 1995 Lunigiana (Italy) earthquake; Cucci and Tertulliani (2011) on the 2009 L‟Aquila (Italy) earthquake (Graham & Kijko, 2006; Hinzen, 2012).

Figure 1.4: Examples of earthquake

rotated objects. Left: obelisk erected in 1850 at Chatak, India, which broke and rotated counter-clockwise during the 1897 Assam earthquake (from Kozák, 2009). Right: dislocated and counter-clockwise rotated tombstones in the Kushiro Cemetery after the 2003-09-26 Tokachi-Oki earthquake (from Hinzen, 2012).

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1.3 Rotational seismic models

Although the existence of rotational motions was discussed from the beginning of the earthquake investigation and studied by Alexander von Humboldt (1769-1859), Charles Lyell (1797-1875) and Charles Darwin (1809-1882), and earthquake induced rotational effects have been observed for centuries, it was necessary to wait until the mid-nineteenth century for the first rotational model (Kozák, 2006; Kozák, 2009; Sargeant & Musson, 2009; Trifunac, 2009; Jaroszewicz et al., 2012).

There are essentially two classes of rotational seismic models: the first class includes two models defined by Robert Mallet in the mid-nineteenth century, named Rot1 and Rot2 by Kozák (2006), which are based on the rotation of bodies to their underlying structures. The second class is derived from advances made in the second half of last century including those in theoretical studies in micromorphic and asymmetric theories of continuum mechanics and in nonlinear physics, improved knowledge of the inner parts of the Earth and seismic waves, and contributions based on modern highly sensitive seismic registration techniques, giving rise to three other types of rotational models, Rot3, Rot4 and Rot5 (Kozák, 2006), which are seismically verified by detailed analyses of the inner focal zone and wave propagation in structured medium; these rotational effects satisfactorily explain observed surface rotations (Kozák, 2009).

To observers in the first half of the nineteenth century, the common explanation of earthquake rotational effects was that there had be an independent vortical movement under the rotated objects responsible for the observed rotations. The lack of specific knowledge of seismic waves did not allow scientists develop models of observed rotational effects until the 1840s, when Robert Mallet, an Irish engineer, presented his mechanical explanation, which provided the first theoretical basis of a physical-mechanical explanation of earthquake rotational effects (Kozák, 2009).

While inspired by many of Lyell‟s studies, Mallet, disagreed with his interpretation of the distortions of the obelisks and in general with the common explanation of earthquake rotational effects by means of so called vortical movements. He proposed that a solid body lying face-to-face on a horizontal underlying plane, subjected to translational wave impact coming in a horizontal direction, may rotate if the point of vertical projection of its center of gravity into the contact plane is not identical with the point of strongest adhesion of the body to its underlay; this is the model named Rot1 by Kozák. Mallet considered another rotational mechanism concerning the rotation of a body in vertical plane, named Rot2 by Kozák, for which a physical body on the surface may gradually be turned in the horizontal plane as a consequence of the fact that incident seismic phases may gradually alter the body‟s horizontal position, especially when the latter wave phases are coming to the considered point under another angle (Ferrari, 2006; Kozák, 2006; Kozák, 2009; Hinzen, 2012). Mallet additionally suggested a method for the determination of the epicenter and the hypocenter of an earthquake, starting from the study of the direction of

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the collapses, the shifting or the rotation of objects, buildings or parts of them (Ferrari, 2006). The explanation of rotations by means of crost vibrations seems first to have been proposed by F. Hoffmann (1838) and later repeated independently by Mallet and others, but did not receive great consideration (Kozák, 2009).

The Rot2 model has been recently confirmed by Hinzen (2012), Yegian et al. (1994) and Athanasopoulos (1995), who concluded that rotations were caused by the twisting moment from the horizontal inertial force that acts on blocks while they are tilting in a rocking motion.

In the decades after the publication of Mallet‟s models, interest in rotational seismic models gradually increased and the rotational effects of individual earthquakes of the time were often reported in the newspapers. More detailed explanation of such phenomena, however, was entirely based on Mallet‟s analysis. After 1900, the analysis of seismic waves and fundamental principles of their propagation were still not sufficiently defined and, moreover there were no instruments capable of detecting rotational motions in either near field and far field. Common explanations of earthquake rotational effects, therefore, did not go beyond the limits of the simple mechanical approach developed by Mallet. During most of the twentieth century, despite developments within the modern theory of seismic wave propagation, the interest amongst seismologists in rotational effects decreased due to the common opinion that they had little to do with the new theories. For example, Gutenberg (1927) wrote that rotational waves were not able propagate, and that even if they were generated at the source, they would be attenuated quickly. Moreover, some experiments which measured vertical ground motion showed that the role of ground tilting in linear-wave motion was usually small (Kozák, 2006; Trifunac, 2009).

However Galitzin (1902), doubting the conclusions based on those experiments, formulated the theory of the transducer response, which he applied simultaneously to tilts and displacements, but found this theory so complicated that in his later work he was forced to neglect the effects of tilts (Trifunac, 2009). Harry F. Reid (1908-1910) observing rotational effects in the 1906 San Francisco earthquake, corrected some weak points in Mallet‟s analysis and pointed out that the observed rotations are too large to be produced by waves of elastic distortion, and that at large epicentral distances, where the shock decreases its energy, rotations were almost negligible, while where the shock was strong, in general for small epicentral distances, they were almost universal. He pointed out that no one observer put the rotatory motion in the early part of the shock, and presented the spectrum of possible configurations of earthquake foci-parameters such as double consequently working sources in one point, two simultaneously active influence of sources in nonidentical points, and the influence of special geometry of undersurface reflecting planes and medium parameters. Until then, Reid‟s analysis represented the most coherent mechanical analysis of rotational effects from earthquakes (Kozák, 2009; Lee, Çelebi et al., 2009; Trifunac, 2009).

Thomas Gray (1880) developed a model, which assumed that ground motion is purely translational, showing that if the vibrations are at right angles to the edge of the rectangular base

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of a column, or along the line joining opposite corners, no rotating moment is developed, while if the shock lies between these directions the column tend to rock on the corner and to rotate around it (Fig. 1.5). Gray‟s model was in total agreement with the laws of mechanics, and the combination of vibrations at right angles offered a simpler explanation for any amount of rotation and for any form of base (Kozák, 2009; Sargeant & Musson, 2009).

Figure 1.5: Gray‟s model showing translational motion

impinging (denoted by the inward pointing arrows) on a

column, in plane view. G is the center of gravity. “If the shock has some intermediate direction, such as e G, and

its effect would be to cause the body to bear heavily on B and at the same time to rotate around B as an axis in the direction of the hands of a watch, i.e. from right to left”. Gray proposed that a shock passing through any of the shaded octants in the figure would cause rotation from right to left, whereas if the shock passed through any of the unshaded octants, rotation from left to right would occur (using this model Meldola & White were not able to constrain the propagation direction) (Sargeant & Musson, 2009).

Meldola and White (1885) accepted that translational motion would be able to produce these rotational effects, as it had been shown by Mallet (1846) and in accordance with Gray‟s model, although they suggested that the fact that the majority of chimneys and other structures were twisted in the same direction provided evidence that translational movement had played a subordinate role only and that a rotational motion of the ground may have been admitted, considering additionally the complicated character of earthquake motion as revealed by seismograph tracings. They also tried to deduce the direction of wave propagation from observations of rotated objects, but they admitted that the propagation direction of whatever motion caused the chimneys to twist was impossible to deduce from the direction of rotation (Sargeant & Musson, 2009).

Later, in 1937 Imamura explained the rotation effects of some objects on the ground surface by the impact body/surface waves, which can cause an object to incline, losing contact in part with the ground surface, and when returning to the vertical some twist occurs with respect to its former position. This theory relates either to differences in the object inertia

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properties or to differences in cohesion/friction between some parts of the object and the ground surface (Teisseyre et al., 2003; Teisseyre et al., 2006).

The mechanism Rot3 relates directly rotation of displacement field both in an inner source near-zone and in the layers beneath a recording station; the effects related to such a mechanism can differ greatly in amplitude and nature, depending on source mechanism and medium structure properties. In the last decades of the twentieth century, taking advantage of advanced continuum mechanics and ray theory and other theoretical achievements on the seismic wave origin and source mechanism, new models related to this mechanism have been proposed, many of them are founded on mutual interference of individual wave phases in the inner seismic source zone (Kozák, 2006).

According to Shimbo (1975, 1995) and Shimbo and Kawaguchi (1976) fracture slip motion causes rotations of the grains adjacent to the fracture plane; thus for small deformation fields, the anti-symmetric part of self-stresses and self-strains becomes related to the proper rotation tensor. For a given symmetric shear stress condition, the rotations of grains due to friction are opposite along perpendicular planes (Fig. 1.6 (left)). However the earthquake process usually proceeds along one slip/fracture plane, the main fault, which produces the asymmetry of rotation and friction stress (Fig. 1.6 (right)) (Teisseyre et al., 2003).

Figure 1.6: Fracture planes and rotation effects due to friction processes or dislocations motions. On the

left, the symmetric case; on the right the fracture along a main fault. The arrows indicate the shear stresses acting along the fracture planes (Teisseyre et al., 2003).

Mechanism Rot4 is linked with the real rotational deformations and the properties of the medium through which the seismic waves propagate. Modern theory of such a medium, usually referred to as a micromorphic medium, was investigated both theoretically and under laboratory conditions, initially by Polish and Japanese seismologists. This theory enables the detection of the rotational component of seismic waves due to wave interaction with the propagation medium treated by the terms of micromorphic description (Teisseyre et al., 2003; Kozák, 2006).

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Mechanism Rot5, recently proposed by Teisseyre (2004), relates to rotation and twist motions (the difference in the rotation effects observed along perpendicular directions on a plane perpendicular to the axis of rotation), and is based on additional constitutional bonds between the antisymmetric part of stresses and density of the self-rotation nuclei (rotating grains) as being related to an internal friction in a homogeneous elastic medium (Teisseyre et al., 2003; Teisseyre, 2009; Kozák, 2006).

Laboratory and observational research in the last two decades in the field of non-linear medium and non-linear physics based on propagation of non-linear deformational waves, performed in cooperation between Russian and Czech seismologists, seems to reveal the existence of a next mechanism, model Rot6, as related to the coherent translation fracture wave (Kozák, 2006).

Recent studies (Tertulliani and Maramai, 1998) show that rotated simple objects may help constrain both the radiation pattern and source mechanism, helping with quantifying such parameters for historic and prehistoric earthquakes (Hinzen, 2012).

In earthquake engineering, the recognition that the rotational components of strong motion contribute significantly to the response of structures started to appear around the 1960s. Many authors in the following decades started to study how the soil-structure interaction affects the response of structures and showed that differential and rotational excitations of flexible, extended, multiple, and separate foundations can lead to large pseudostatic shears and moments in structural bodies, additionally pointing out how rocking excitation becomes significant for tall structures supported by soft soil deposits, while torsional excitations can dominate in the response of long and stiff structures supported by soft soils. Furthermore, there are many examples indicating enormous increase of rotation effects caused by consecutive impacts of seismic body and surface waves (Teisseyre et al., 2006; Trifunac, 2006; Smerzini, 2008; Trifunac, 2009; Falamarz-Sheikhabadi & Ghafory-Ashtiany, 2012).

1.4 Measurements of rotational ground motion

Droste and Teisseyre (1976), using an azimuthal array of six horizontal seismographs oriented in six directions differing by 30°, derived rotational seismograms of rock bursts from a nearby mine (Teisseyre et al., 2003; Lee, Çelebi et al., 2009), and were the first to record rotational waves successfully in Poland (Lee, Huang et al., 2009).

The early efforts also included studies of explosions, such as the works of Graizer (1991), who recorded tilts and translational motions in the near field of two nuclear explosions using seismological observatory sensors to directly measure point rotations, and Nigbor (1994), who directly measured rotational and translational ground motions and observed significant amounts of near-field rotational motions (660 μrad at 1 km distance) near a large chemical explosion at

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the Nevada Test Site (NTS) using a commercial rotational sensor (GyroChip) (Lee, Çelebi et al., 2009; Lee, Huang et al., 2009; Lin et al., 2009).

Takeo (1998) described and analysed three translational and three rotational components of ground velocity recorded during two nearby earthquakes offshore of Ito on the Izu peninsula, Japan, in 1997, measuring rotational motions with Systron Donner triaxial gyro sensors with a full-scale output capacity of _{(Trifunac, 2006; Takeo, 2009).}

The strong-motion observation in Taiwan (Lin et al., 2009) and Japan showed that the amplitudes of rotations can be one to two orders of magnitude greater than expected from the classical linear theory, as first noted by Takeo 1998. Theoretical work has further suggested that, in granular, cracked, or biologically disturbed, often dry continua, asymmetries of the stress and strain fields can create rotations in addition to those predicted by the classical elastodynamic theory for a perfect continuum (Lee et al., 2007; Lee et al., 2012). Lin et al. (2011) carried out the first teleseismic six-degree-of-freedom measurements (for the 2008 Mw 7.9 Wenchuan (Sichuan,

China) earthquake) including three components of rotational motions recorded by a commercial rotation-rate sensor (model R-1, made by eentec) and three components of translational motions recorded by a traditional seismometer (STS-2) at the NACB station in Taiwan, showing consistent observations in waveforms of rotational motions and translational motions in sections of Rayleigh and Love waves, and also exploiting additional information and finding significant errors in the translational records of the telesismic surface waves due to the sensitivity of inertial translational sensors to horizontal rotational motions. In particular, they observed in the section of Love waves significant amplitudes for radial rotational motions, in contradiction with previous studies which stated that horizontally polarized Love waves generate rotations around a vertical axis only (Igel

et al., 2005). This observation was perhaps mainly due to the incident angle of the trapped SH

waves, as indicated by Eq. (A.18) (see Appendix A).

Average rotational motions can be approximated from the differences in the recordings of accelerometer arrays using methods valid for seismic waves having wavelengths that are long compared to the distances between sensors (e.g. Trifunac, 1979; 1982; Oliveira and Bolt, 1989; Spudich et al., 1995; Bodin et al., 1997; Huang, 2003; Suryanto et al., 2006; Wassermann et al., 2009) (Lee et al., 2012), and have been also estimated theoretically, using kinematic source models (e.g., Bouchon & Aki, 1982; Wang et al., 2009) and linear elastodynamic theory of wave propagation in elastic solids (Lee and Trifunac, 1985; 1987) (Lee et al., 2012).

An alternative approach to evaluate the importance of the rotational motions is to make use of numerical and mathematical models that allow the simulation of seismic phenomena (Li et al. (2004)), including, for example, computing the surface rotation field along a topographical profile of an elastic half-space or layered media subjected to the incidence of elastic waves by using different numerical methods (e.g., Godinho et al. (2009) who pointed out the effects of topography on rotational ground motion in both frequency and time domains).