When the seismic wave propagates in the medium, the motion of the particle usually includes the translation, rotation and deformation (Lee W.H.K. et al., 2009; Trifunac M.D., 2009).Traditional seismology focuses on observing and simulating the three-component translational motion of the earth surface and buildings. However, a complete representation of the ground motion induced by earthquakes consists not only of those three components of translational motion, but also the three components of rotational motion and six components of strain (Aki K. et al., 2002). Although the rotation effect caused by earthquake has been observed for more than one hundred years, the lack of suitable instruments has contributed to certain difficulties in measuring rotational motion, and it is generally believed that rotational motion is very weak, and therefore not much attention has been paid to its research. Modern observational seismology is still dominated by measuring translational motion.
Recently, strong earthquakes have occurred frequently, resulting in heavy casualties and economic losses. Earthquake prediction research currently faces unprecedented opportunities and challenges. Gu Haoding et al. (1988) pointed out that rotational seismology not only has theoretical significance, but will also promote the in-depth study of earthquake prediction. In the past few decades, with further development of seismic wave rotation theory and the emergence of precision instruments, much progress has been made in rotational seismology (Takeo M., 1998; Trifunac M.D. et al., 2001; Teisseyre R. et al., 2003; Huang B.S. et al., 2006). It has been shown that rotational observational studies provide additional wave field information, such as phase velocity, direction of propagation, and anisotropy (Takeo M. et al., 1997; Igel H. et al., 2005, 2007; Suryanto W. et al., 2006). Combining rotational motion with translational motion can provide more information on the Earth's structure, which makes it possible to perform tomography without travel-time parameters and provides more detailed information on the process of earthquake ruptures. In this paper, the application of rotational motion in seismology is described in order to deepen the understanding of rotational seismology and to promote the development of rotational seismology in China.
1 ROTATIONAL SEISMIC TOMOGRAPHYSeismic source and structure have always been the main content of classical seismology research. Seismic tomography is the main method for studying underground structures. Highly resolved tomographic images are essential for a variety of applications, ranging from global-scale geodynamics to local-scale engineering and exploration. There are two general strategies to improve seismic tomography: ① developing more elaborate modeling and inversion techniques, such as finite-frequency tomography (Yoshizawa K. et al., 2004) or nonlinear full-waveform inversion (Fichtner A. et al., 2008); ② incorporating previously unexploited data such as surface-wave amplitudes, exotic phases, and measurements of seismically induced strain or rotations.
Traditional tomographic methods rely on seismic wave travel time whether they use body waves or surface waves. The determination of frequency-dependent surface-wave phase velocities has, for a long time, been one of the most important tools in determining 3-D seismic velocity structure on both a regional and global scale (Nataf H.C. et al., 1984; Snieder R., 1988a, 1988b). On a small scale, near-surface low-velocity structures crucial for the estimation of hazard-relevant site effects can be determined using ambient noise measurements (Kind F. et al., 2005; Milana G. et al., 1996). The discovery and measurement of seismic rotational motion components enrich the seismic observation information, making it possible to measure the surface wave phase velocity in the case of a single seismic station, and open a new chapter of seismic tomography without the need for travel time parameters.
1.1 Teleseismic Rayleigh Wave TomographyIn recent years, ground rotation has received more and more attention from seismologists. One of the key questions about rotational motion is what information can we get from the rotational motion excited by an earthquake? It is found that the rotational component around the vertical axis is only sensitive to seismic SH waves, but not sensitive to P-SV waves, so we can get accurate data of SH wave arrival time (Takeo M. et al., 1997). Igel H. et al., (2005) suggested that the transverse acceleration and vertical rotation-rate should be in phase with relative amplitude scaled by two times the horizontal phase velocity. Suryanto W., (2006) also proposed that radial acceleration and the transverse rotation-angle should be in phase with relative amplitude scaled by phase velocity multiplied by frequency, detailed observations of horizontal axis rotational motions are lacking at teleseismic distances.
Lin C.J. et al., (2011) report the findings for the teleseismic six degree-of-freedom measurements (including three components of rotational motions and three components of translational motions recorded by a rotational-rate sensor), accompanied by a theoretical formula of the Rayleigh wave phase velocity in a simple semi-infinite Poisson medium. For a simple half-space Poisson solid, the Rayleigh wave displacements propagating along the X-axis and Z-axis at zero depth are given by Lay T. et al., (1995) as
$ \begin{align} &{{u}_{X}}=-0.42Ak\text{sin}(\omega t-kx) \\ &{{u}_{Z}}=0.62Ak\text{cos}(\omega t-kx) \\ \end{align} $ | (1) |
Where A is the P-wave amplitude, ω is the angular frequency, x is distance, and k is the Rayleigh wave number. We refer to the X-axis and Z-axis directions as radial and vertical. In addition, the Z-axis is positive in the downward direction. The velocity
$ \begin{align} &{{{\dot{u}}}_{X}}=-0.42Ak\omega \cos \left(\omega t-kx \right) \\ &{{{\ddot{u}}}_{X}}=0.42Ak{{\omega }^{2}}\text{sin}\left(\omega t-kx \right) \\ &{{{\ddot{u}}}_{Z}}=-0.62Ak{{\omega }^{2}}\text{cos}\left(\omega t-kx \right) \\ \end{align} $ | (2) |
Accordingly, the rotation angle θ_{Y} and rotation rate
$\begin{align} &{{\theta }_{Y}}=-\frac{\partial {{u}_{Z}}}{{{\partial }_{X}}}=-0.62A{{k}^{2}}\text{sin}(\omega t-kx) \\ &{{{\dot{\theta }}}_{Y}}=-0.62A{{k}^{2}}\omega \text{cos}(\omega t-kx) \\ \end{align} $ | (3) |
By comparing the acceleration of the translational motion along the Z-axis and the rotation rate around the Y-axis, we can obtain the theoretical formula for calculating the phase velocity
$ \frac{{{{\ddot{u}}}_{Z}}}{{{{\dot{\theta }}}_{Y}}}~=c $ | (4) |
where c is the apparent phase velocity of Rayleigh wave.The results show that the Z-axis acceleration recorded by the teleseismic Rayleigh wave is consistent with the rotation rate of the horizontal Y-axis in phase, and the amplitude ratio between them equals the phase velocity.
According to formula (4), if only the fundamental mode of the Rayleigh wave is considered, we can theoretically deduce the local Rayleigh wave dispersion curve by simultaneously measuring the lateral rotational rate and the vertical translational acceleration of a single station. The phase velocity structure information is derived from the dispersion curve.
1.2 Teleseismic Love Wave TomographySince the 20th century, the advancement of ring laser technology has promoted the development of observation instruments. More and more rotational motions induced by distant and regional earthquakes have been recorded (Takeo M., 1998; Teisseyre R. et al., 2003; Huang B. S. et al., 2006; Suryanto W. et al., 2006; Igel H. et al., 2007). Igel H. et al., (2005) reported the first consistent observations of rotational motions around a vertical axis induced by distant large earthquakes. It shows that the rotations observed are consistent in waveform and amplitude with collocated recordings of transverse accelerations recorded by a standard seismometer.
In order to compare translations with the vertical component of the vector of rotation, the horizontal components were rotated into radial and transverse directions. As the Rayleigh wave cannot produce a vertical rotational motion, Love waves are horizontally polarized hence generating rotations around a vertical axis only. Assuming that the plane wave displacement of the transversely polarized plane wave can be expressed as
$u=[0, {{u}_{Y}}\left(t-x/c \right), 0] $ | (5) |
Where c is the horizontal phase velocity. The vector of rotation (curl) is thus given as
$ \frac{1}{2}\nabla \times u=[0, 0, -\frac{1}{2c}{{{\dot{u}}}_{_{Y}}}\left(t-x/c \right)] $ | (6) |
Accordingly, the rotation rate around the Z-axis can be expressed as
$ {{\Omega }_{Z}}\left(x, t \right)=-\frac{1}{2c}{{{\ddot{u}}}_{Y}}\left(t-x/c \right) $ | (7) |
The above equations imply that under the given conditions, the rotation rate and transverse acceleration are in phase and the amplitudes are related by the following relationship,
$ {{{\ddot{u}}}_{_{Y}}}\left(x, t \right)/{{\Omega }_{Z}}\left(x, t \right)=-2c $ | (8) |
It is found that for the events with the rotation motions recorded, the waveform similarity between the transverse acceleration and the vertical rotation rate is significant. This similarity is not only limited to the most dominant Love wave signals but the fit continues long after the fundamental mode surface waves have passed. The time-dependent similarity between the rotation rate and transverse acceleration can be quantified by sliding a time window of appropriate length (longer than one dominant period) along the time-series and the zero-lag normalized cross-correlation coefficient that is defined between 0 (no similarity) and 1 (perfect match) can be calculated. In fact, the phase velocity can be estimated by selecting the appropriate length of the sliding window to best fit the waveform of the seismogram (Fig. 1). This method can also be used in the frequency domain, and the phase velocity can be estimated by the spectrum ratio of transverse acceleration to rotation velocity. It can be seen that the main period of wave decreases with time due to the dispersion effect of surface wave as is shown in Fig. 1(a). Accordingly, the phase velocity obtained by fitting the sliding time window decreases gradually with the increase of time, also as shown by Fig. 1(b).
In a theoretical study, Ferreira A.M.G. et al., (2009) showed that the relation between horizontal translation and vertical rotation also holds for fundamental Love waves in smooth, laterally heterogeneous media. Using Full-Ray Theory (FRT) simulations, Ferreira A.M.G. et al., (2009) found that rotation amplitudes are strongly influenced by the local structure near the receiver. The FRT calculation can help to determine the dispersion curve of the Love waves, and then estimate the local 1-D shear wave velocity structure under the seismic station from the collocated point measurements of the rotation rate and the transverse acceleration of the ground motion. In addition to phase velocities, it is important to know that information on the direction of propagation is contained in the back azimuth-dependent phase fit between rotation and translations. Supposing that the back azimuth is unknown, by seeking the propagation direction that maximizes the correlation coefficient in sliding windows of appropriate length along the whole seismogram, the back azimuth can be estimated to be within a few degrees (Igel H. et al., 2007; Pham N. D. et al., 2009a; Hadziioannou C. et al., 2012).
Both the teleseismic Rayleigh wave and the Love wave tomography method show that the rotational motion of seismic waves plays an increasingly important role in seismological research. Considering different limited conditions, there is a certain proportional relationship between the different components of the translational motion and the rotational motion, and the ratio is scaled to the phase velocity. The estimation of the horizontal phase velocity would thus be based upon a point measurement of rotation and translation motions, otherwise, it can only be estimated by an array of seismometers or additional strain measurements. This single-point measurement using a rotating sensor is ideal for the observation of sparse or single station area, and does not require travel time information, which provides a new idea for seismic tomography.
2 AMBIENT NOISE AND P CODA WAVESContinuously improving sensitivity of the ring laser has recently made it possible to detect rotational ground motions induced by microseisms (including ambient noise caused by the interaction of the atmosphere, ocean waves and the ground). It is shown that Rayleigh wave dispersion curves can be derived by correlating long time series of ambient noise (micro-seismicity) and that the velocity structure thus derived can be used to image 3-D structures (Campillo M. et al., 2003; Shapiro N.M. et al., 2004, 2005). Evaluating amplitude ratios of collocated observations of translational and rotational motions, Hadziioannou C. et al., (2012) found a consistent high correlation between rotation rate and transverse acceleration signals in the secondary microseismic band (0.1Hz-0.2Hz), which is not present at other frequencies. In their study, they recovered local phase velocities as well as the back azimuth of oceanic noise sources (Fig. 2). The back azimuth of the Tohoku-Oki earthquake obtained by the sliding window cross-correlation analysis is 37.1°, which is perfectly consistent with the actual position of 37° between them. The cross-correlation coefficient for back azimuth continues to be large at t 300° (outside the region of the Tohoku-Oki earthquake and its aftershocks), which suggests that this may be the location of a fixed source of ocean noise. The amplitude ratio of transverse acceleration and vertical rotation rate gives the local phase velocities (Hadziioannou C. et al., 2012). As the ring laser only records Love waves, it helps distinguish ambient noise and surface wave (Love wave) energy in the ocean-generated noise field.
The partitioning of P and S energy and the stabilization of the ratio between the two are an important constraint on the scattering properties of the medium. As a ring laser is sensitive to SH-type motion only, one of the expectations of ring laser measurements is to help seismologists separating P and S waves in the wave field (Takeo M. et al., 1997). Previous studies (Igel H. et al., 2007) have suggested that the rotational motion component of the P coda wave in the seismogram around the vertical axis is of great significance. Theoretically, in spherically symmetric isotropic media, P-SV waves do not produce a rotation component around the vertical axis, so we should not observe rotational signals around the vertical axis before the onset of SH waves. It was a surprise to discover considerable rotational energy in a time window containing the P-coda in the vertical rotation rate recordings by the ring laser in Wettzell (Igel H. et al., 2007; Pham N. D. et al., 2009a), as shown in Fig. 3. It shows that with an increasing cutoff period, the amplitude of the P-coda rotations is decreased. Amplitudes of the P-coda rotational signals decrease significantly for a cutoff period of T=5s and almost disappear when the cutoff period reaches a value of 10s. This implies that the energy of the observed P-coda rotations is predominant at high frequencies. This phenomenon is observed for all events with visible P-coda rotations.
There are several possible explanations for the phenomenon of P-coda rotations: ① tilt-ring laser coupling, when P waves arrive, they do generate tilts at the earth's surface (i.e., rotational motions around horizontal axes). This will contaminate the ring laser measurements through changes of the surface normal with respect to the Earth's rotation axis and may potentially contribute to the phenomenon of rotations in the P coda (Schreiber U. et al., 2006; Igel H. et al., 2007). ② P-SH scattering in the crust, ③ topographic scattering, and ④ anisotropy are the additional possible explanations.
The tilt-ring laser coupling is assumed to be small in the far-field case (McLeod D.P. et al., 1998; Igel H. et al., 2007), and is usually negligible. Detailed and further systematical investigations can be seen in a companion article (Pham N.D. et al., 2009b). The contributions of topographic scattering and anisotropy on the P-coda rotations are expected to be small but were never properly quantified. Significant rotations around the vertical axis in the P coda of teleseismic signals are either directly visible or can be inferred through the investigation of cross correlations between the transverse component (the component that is perpendicular to the great circle connecting the earthquake and the seismometer) of translation acceleration and ring laser rotation rate. Pham N.D. et al., (2009a) showed that the observed signals can be explained by the P-SH scattering of high-frequency seismic waves in the three-dimensional random crust through detailed signal analysis and simulation studies of wave propagation in three-dimensional random media.
3 OBSERVATION OF EARTH'S FREE OSCILLATIONGiant earthquakes radiate mechanical energy into and around the earth leading to worldwide oscillatory ground displacements (Park J. et al., 2005). As a global standing wave mode formed after constructive interference of seismic wave fields, the earth's free oscillations (or normal modes) are characterized by discrete frequencies that depend primarily on the motion type (toroidal or spheroidal modes) and the structure of our planet. This implies that observations of free oscillations provide some of the most important large-scale constraints on a variety of elastic parameters, attenuation, and density of the earth's deep interior (Gilbert F. et al., 1975; Ishii M. et al., 1999; Beghein C. et al., 2003).
Large earthquakes excite free oscillations significantly above our planet's background noise level, however, because of the high noise level on the horizontal translational components in classical seismometers, toroidal modes (i.e. the superposition of Love waves) have lower Q (higher attenuation) than spheroidal modes, so toroidal oscillations decay more rapidly. Long-period toroidal modes, which correspond to a superposition of SH-type motions, have always been among the most challenging observations in seismology (Widmer-Schnidrig R. et al., 2009). Nevertheless, given sufficient sensitivity, rotational sensors can be used to observe toroidal modes directly by recording motions around a locally vertical axis with very low (theoretically zero) cross sensitivity to translational motions. Igel H. et al., (2011) reported the first measurements of the toroidal eigenmodes of the earth by a rotation sensor of the ring laser at Wettzell. The observed and synthetic amplitude spectra of transverse and rotational motions are calculated, as shown in Fig. 4. Whether it is a transverse seismic record or the vertical rotational records of ring laser, the main toroidal modes of earth's free oscillation can be clearly distinguished.
Comparing the first observations of free oscillations by a ring laser with synthetic seismograms calculated by normal mode summation in a spherically symmetric, nonrotating, perfectly elastic, isotropic Earth model, Nader M.F. et al., (2012) found that the vertical axis rotational spectra contained also the energy from spheroidal oscillations. Investigating this cross-coupling of modal types in a subsequent study, Nader M.F. et al., (2015) concluded that the observed spheroidal peaks can be explained by the effects of the earth's daily rotation, its hydrostatic ellipticity, and structural heterogeneity.
4 JOINT INVERSION OF SEISMIC TRANSLATION AND ROTATIONThe ground rotation caused by the earthquake contains information about the earth's structure. The possibility of the derivation of local dispersion relations combining translations and rotations leads to the question of which subsurface volume one actually "sees" and down to what depth velocity perturbations can be recovered. The method of choice to answer these types of questions is the adjoint method (Tromp J. et al., 2005; Fichtner A. et al., 2006), with which volumetric sensitivity of the observables (mostly travel times) to structural perturbations can be calculated. Fichtner A. et al., (2009) introduced a new observable quantity-apparent shear wave velocity β_{a}. They define the apparent shear wave speed in terms of the rms amplitudes of the filtered and windowed velocity and rotation signals,
${{\beta }_{a}}\left({{X}^{r}} \right)=\frac{1}{2}\frac{{{A}_{v}}\left({{X}^{r}} \right)}{{{A}_{\omega }}\left({{X}^{r}} \right)} $ | (9) |
Where A_{v}(X^{r}) and A_{ω}(X^{r}) represent the root-mean-square amplitude of the translational velocity record and the root-mean-square amplitude of the rotation signal, respectively, which are defined as,
$ \begin{align} &{{A}_{v}}=\sqrt{\int_{\text{R}}{{{\left[ F\times \left(Wv \right) \right]}^{2}}\text{d}\mathit{t}}} \\ &{{A}_{w}}=\sqrt{\int_{\text{R}}{{{\left[ F\times \left(Ww \right) \right]}^{2}}\text{d}\mathit{t}}} \\ \end{align} $ | (10) |
The symbols F is the convolution filter.W is the time wirdow to use.v is the translational speed record; ω is the rotational motion record.
The result of the apparent shear wave velocity is that a new sensitive kernel appears. The results based on analytical solutions and numerical simulation show that the sensitive kernel of β_{a} has the largest absolute value only near the receiver, not around the source. That is, the sensitivity near the source vanishes, along with high sensitivity in the vicinity of the receiver (Fig. 5). It has been confirmed in the study of the shear wave (S-wave) and surface wave (Rayleigh wave and Love wave). Fichtner A. et al., (2009) confirmed that the measurement of shear wave velocity can theoretically be used for shear wave velocity imaging in a single station area. Inspired by Fichtner A. et al., (2009), using synthetic data, Bernauer M. et al., (2009) implemented local cross-hole tomography and regional-scale earthquake tomography. The results indicate that S-wave speed variations can be retrieved accurately from collocated rotation and translation measurements (i.e. apparent shear wave velocity). This implies that a tomographic inversion scheme for near-receiver structures based on rotation and translations is conceivable and further highlights the potential of additional rotation measurements. Bernauer M. et al., (2009) illustrated the preliminary results of translational velocity and apparent shear wave velocity tomography inversion using synthetic data (Fig. 6). The inversion of the apparent shear wave velocity reduces the shallow uncertainty and improves the vertical resolution of regional structure imaging, compared with the inversion using root mean square velocity information alone. Using the amplitude information of the translational motion and rotation signals in the tomography method with apparent shear wave velocities, the trade-off between the focal depth and the structure can be reduced without the seismic wave travel time information. The measurement of rotation component of ground motion can be used as a supplement to the standard tomography method, which can restrict and extend the traditional tomography method.
The kinematics of earthquake sources, i.e., the evolution of the rupture process in space and time on a finite fault, are the key to our understanding of seismic hazard and earthquake dynamics. The inverse problem of inferring a kinematic source model from seismic data is ill posed, with solutions critically depending on the available data, the parameterization of the model and the inversion and regularization strategy. A series of studies have shown that measuring the rotational motion of the ground has great potential for extracting information about seismic source parameters (Bouchon M. et al., 1982; Takeo M. et al., 1997; Takeo M., 1998; Wang Haijiang et al., 2009). Using the fault and receiver geometry of the Tottori 2000 earthquake as a test bed, based on the synthetic Bayesian finite source inversion method, Bernauer M. et al., (2014) have conducted a simulation experiment, and event geometry and stations are shown in Fig. 7. In scenario Ⅰ, only translational velocity recordings are used. In scenario Ⅱ, half of the velocity recordings are randomly replaced by rotation recordings, thus keeping the total amount of data constant. The results indicate that the incorporation of rotational ground motion recordings can significantly reduce non-uniqueness and trade-offs in finite source inversions, provided that measurement uncertainties are similar to or below the uncertainties of translational velocity recordings. The availability of six-component (6-C) seismic arrays, i.e., arrays that provide three components of translation (3-C) and three components of rotation data, can increase the amount of data available for the inversion problem, better constraining the source parameters and the source rupture process.
Rotational seismology is an emerging field for studying all aspects of rotational ground motions induced by earthquakes, explosions, and ambient vibrations. It is of interest to a wide range of geophysical disciplines, including strong-motion seismology, broadband seismology, earthquake engineering, earthquake physics, seismic instrumentation, seismic hazards, seismotectonics, and geodesy. Collocated measurements of translations and rotations in seismology provide a new idea for the development of seismic tomography methods—rotational seismic tomography. Further, the introduction of apparent shear wave velocity can reduce the trade-offs between the source depth and structure, improve the vertical resolution of the region structure, and constrain and extend the traditional tomographic results. High-resolution ring laser observations can be used to measure ultra-low frequency (less than 5mHz) seismic signals caused by Earth free oscillation and the partitioning of P and S energy. The collocated measurement of seismic translational motion and rotational motion greatly improves the available amount of inversion data, and plays a significant role in reducing the non-uniqueness of seismic multi-parameter inversion and improving resolution. As a new subject, rotational seismology has a broad prospect in the fields of building safety monitoring and protection in engineering earthquakes, gravitational wave detection in astrophysics and geophysical exploration.
Although many earthquake-induced rotational motion phenomena can be observed by high-resolution ring laser technology, and the multi-component observations allow the estimate of wavefield properties (e.g. phase velocities, propagation direction) that otherwise are only accessible with high accuracy through array (or to some extent additional strain) measurements. However, the observation of rotational motion relies on the large ring laser (G-ring) and still lacks a portable, highly sensitive rotary motion sensor. In addition, the phase velocities estimated by taking the ratio of transverse acceleration and rotation rate, based on the assumption of the plane transverse wave propagation theory, what are the consequences of non-planar wave fronts and the inclusion of dispersive effects? How can we remove the effects of tilt (i.e. the rotational component around horizontal axis) from conventional seismic records? What is the influence of seismic anisotropy on rotational measurements? Many of these issues remain to be resolved. To answer these questions, a more in-depth study of the theory of rotational seismology, observation instruments, data processing analysis methods, wave propagation theory will be necessary, so as to fully describe the particle motion (including translational motion, rotational motion and deformation), and then, explore the earth on which we live with more precise models and theory.
ACKNOWLEDGMENTSMany thanks to the two reviewers for their meticulous comments and valuable suggestions.
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