Definition of the Subject
When a sudden rupture occurs in the Earth, elastic (seismic) waves are generated. When these waves reach the Earth's surface, we may feel themas a series of vibrations, which we call an earthquake. Seismology is derived from the Greek word \( { \sigma \varepsilon \iota \sigma \mu \acute{o}\varsigma } \)(seismos or earthquake) and \( { \lambda \acute{o} \gamma o\varsigma }\) (logos or discourse); thus, it is the science of earthquakes and related phenomena. Seismic waves can be generatednaturally by earthquakes or artificially by explosions or other means. We define earthquake monitoring as a branch of seismology, whichsystematically observes earthquakes with instruments over a long period of time.
Instrumental recordings of earthquakes have been made since the later part of the 19th century by seismographic stations and networks of varioussizes from local to global scales. The observed data have been used, for example, (1) to compute the source parameters of...
Abbreviations
 Active fault:

A fault (q.v.) that has moved in historic (e. g., past 10,000 years) or recent geological time (e. g., past 500,000 years).
 Body waves:

Waves which propagate through the interior of a body. For the Earth, there are two types of seismic body waves: (1) compressional or longitudinal (P wave), and (2) shear or transverse (S wave).
 Coda waves:

Waves which are recorded on a seismogram (q.v.) after the passage of body waves (q.v.) and surface waves (q.v.). They are thought to be back‐scattered waves due to the Earth's inhomogeneities.
 Earthquake early warning system (EEWS):

An earthquake monitoring system that is capable of issuing warning message after an earthquake occurred and before strong ground shaking begins.
 Earthquake precursor:

Anomalous phenomenon preceding an earthquake.
 Earthquake prediction:

A statement, in advance of the event, of the time, location, and magnitude (q.v.) of a future earthquake.
 Epicenter:

The point on the Earth's surface vertically above the hypocenter (q.v.).
 Far‐field:

Observations made at large distances from the hypocenter (q.v.), compared to the wave‐length and/or the source dimension.
 Fault:

A fracture or fracture zone in the Earth along which the two sides have been displaced relative to one another parallel to the fracture.
 Fault slip:

The relative displacement of points on opposite sides of a fault (q.v.), measured on the fault surface.
 Focal mechanism:

A description of the orientation and sense of slip on the causative fault plane derived from analysis of seismic waves (q.v.).
 Hypocenter:

Point in the Earth where the rupture of the rocks originates during an earthquake and seismic waves (q.v.) begin to radiate. Its position is usually determined from arrival times of seismic waves (q.v.) recorded by seismographs (q.v.).
 Intensity, earthquake:

Rating of the effects of earthquake vibrations at a specific place. Intensity can be estimated from instrumental measurements, however, it is formally a rating assigned by an observer of these effects using a descriptive scale. Intensity grades are commonly given in Roman numerals (in the case of the Modified Mercalli Intensity Scale, from I for “not perceptible” to XII for “total destruction”).
 Magnitude, earthquake:

Quantity intended to measure the size of earthquake at its source, independent of the place of observation. Richter magnitude (\( { M_\mathrm{L} } \)) was originally defined in 1935 as the logarithm of the maximum amplitude of seismic waves in a seismogram written by a Wood–Anderson seismograph (corrected to) a distance of 100 km from the epicenter. Many types of magnitudes exist, such as body‐wave magnitude (\( { m_\mathrm{b} } \)), surface‐wave magnitude (\( { M_\mathrm{S} } \)), and moment magnitude (\( { M_\mathrm{W} } \)).
 Moment tensor:

A symmetric second‐order tensor that characterizes an internal seismic point source completely. For a finite source, it represents a point source approximation and can be determined from the analysis of seismic waves (q.v.) whose wavelengths are much greater than the source dimensions.
 Near‐field:

A term for the area near the causative rupture of an earthquake, often taken as extending a distance from the rupture equal to its length. It is also used to specify a distance to a seismic source comparable or shorter than the wavelength concerned. In engineering applications, near‐field is often defined as the area within 25 km of the fault rupture.
 Plate tectonics:

A theory of global tectonics (q.v.) in which the Earth's lithosphere is divided into a number of essentially rigid plates. These plates are in relative motion, causing earthquakes and deformation along the plate boundaries and adjacent regions.
 Probabilistic seismic hazard analysis:

Available information on earthquake sources in a given region is combined with theoretical and empirical relations among earthquake magnitude (q.v.), distance from the source, and local site conditions to evaluate the exceedance probability of a certain ground motion parameter, such as the peak ground acceleration, at a given site during a prescribed time period.
 Seismic hazard:

Any physical phenomena associated with an earthquake (e. g., ground motion, ground failure, liquefaction, and tsunami) and their effects on land use, man‐made structure, and socio‐economic systems that have the potential to produce a loss.
 Seismic hazard analysis:

The calculation of the seismic hazard (q.v.), expressed in probabilistic terms (See probabilistic seismic hazard analysis, q.v.). The result is usually displayed in a seismic hazard map (q.v.).
 Seismic hazard map:

A map showing contours of a specified ground‐motion parameter or response spectrum ordinate for a given probabilistic seismic hazard analysis (q.v.) or return period.
 Seismic moment:

The magnitude of the component couple of the double couple that is the point force system equivalent to a fault slip (q.v.) in an isotropic elastic body. It is equal to rigidity times the fault slip integrated over the fault plane. It can be estimated from the far‐field seismic spectrum at wave lengths much longer than the source size. It can also be estimated from the near‐field seismic, geologic and geodetic data. Also called “scalar seismic moment” to distinguish it from moment tensor (q.v.).
 Seismic risk:

The risk to life and property from earthquakes.
 Seismic wave:

A general term for waves generated by earthquakes or explosions. There are many types of seismic waves. The principle ones are body waves (q.v.), surface waves (q.v.), and coda waves (q.v.).
 Seismograph:

Instrument which detects and records ground motion (and especially vibrations due to earthquakes) along with timing information. It consists of a seismometer (q.v.) a precise timing device, and a recording unit (often including telemetry).
 Seismogram:

Record of ground motions made by a seismograph (q.v.).
 Seismometer:

Inertial sensor which responds to ground motions and produces a signal that can be recorded.
 Source parameters of an earthquake:

The parameters specified for an earthquake source depends on the assumed earthquake model. They are origin time, hypocenter (q.v.), magnitude (q.v.), focal mechanism (q.v.), and moment tensor (q.v.) for a point source model. They include fault geometry, rupture velocity, stress drop, slip distribution, etc. for a finite fault model.
 Surface waves:

Waves which propagate along the surface of a body or along a subsurface interface. For the Earth, there are two common types of seismic surface waves: Rayleigh waves and Love waves (both named after their discoverers).
 Tectonics:

Branch of Earth science which deals with the structure, evolution, and relative motion of the outer part of the Earth, the lithosphere. The lithosphere includes the Earth's crust and part of the Earth's upper mantle and averages about 100 km thick. See plate tectonics (q.v.).
 Teleseism:

An earthquake at an epicentral distance greater than about 20° or 2000 km from the place of observation.
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Acknowledgments
We thank John Evans, Fred Klein, Woody Savage, and Chris Stephens for reviewing the manuscript, their comments and suggestionsgreatly improved it. We are grateful to Lind Gee and Bob Hutt for information about the Global Seismographic Network (GSN) and for providinga high‐resolution graphic file of an up‐to‐date GSN station map.
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Appendix: A Progress Report on Rotational Seismology
Appendix: A Progress Report on Rotational Seismology
Seismology is based primarily on the observation and modeling of three orthogonal components of translational ground motions. Although effects of rotational motions due to earthquakes have long been observed (e. g., [80]), Richter (see, p. 213 in [97]) stated that:
Perfectly general motion would also involve rotations about three perpendicular axes, and three more instruments for these. Theory indicates, and observation confirms, that such rotations are negligible.
However, Richter provided no references for this claim, and the available instruments at that time did not have the sensitivity to measure the very small rotation motions that the classical elasticity theory predicts.
Some theoretical seismologists (e. g., [4,5]) and earthquake engineers have argued for decades that the rotational part of ground motions should also be recorded. It is well known that standard seismometers and accelerometers are profoundly sensitive to rotations, particularly tilt, and therefore subject to rotation‐induced errors (see e. g., [39,40,41,93]). The paucity of instrumental observations of rotational ground motions is mainly the result of the fact that, until recently, the rotational sensors did not have sufficient resolution to measure small rotational motions due to earthquakes.
Measurement of rotational motions has implications for: (1) recovering the complete ground‐displacement history from seismometer recordings; (2) further constraining earthquake rupture properties; (3) extracting information about subsurface properties; and (4) providing additional ground motion information to engineers for seismic design.
In this Appendix, we will first briefly review elastic wave propagation that is based on the linear elasticity theory of simple homogeneous materials under infinitesimal strain. This theory was developed mostly in the early nineteenth century: the differential equations of the linear elastic theory were first derived by Louis Navier in 1821, and Augustin Cauchy gave his formulation in 1822 that remains virtually unchanged to the present day [103]. From this theory, Simeon Poisson demonstrated in 1828 the existence of longitudinal and transverse elastic waves, and in 1885, Lord Rayleigh confirmed the existence of elastic surface waves. George Green put this theory on a physical basis by introducing the concept of strain energy, and, in 1837, derived the basic equations of elasticity from the principle of energy conservation. In 1897, Richard Oldham first identified these three types of waves in seismograms, and linear elasticity theory has been embedded in seismology ever since.
In the following we summarize recent progress in rotational seismology and the need to include measurements of rotational ground motions in earthquake monitoring. The monograph by Teisseyre et al. [109] provides a useful summary of rotational seismology.
Elastic Wave Propagation
The equations of motion for a homogeneous, isotropic, and initially unstressed elastic body may be obtained using the conservation principles of continuum mechanics (e. g., [30]) as
and
where θ is the dilatation, ρ is the density, u _{ i } is the ith component of the displacement vector \( { \vec{u} } \), t is the time, and λ and μ are the elastic constants of the media. Eq. (A1) may be rewritten in vector form as
If we differentiate both sides of Eq. (A1) with respect to x _{ i }, sum over the three components, and bring ρ to the right‐hand side, we obtain
If we apply the curl operator (\( { \nabla\times } \)) to both sides of Eq. (A3), and note that
we obtain
Now Eqs. (A4) and (A6) are in the form of the classical wave equation
where Ψ is the wave potential, and v is the wave‐propagation velocity (a pseudovector; wave slowness is a proper vector). Thus a dilatational disturbance θ (or a compressional wave) may be transmitted through a homogenous elastic body with a velocity \( { V_\mathrm{P} } \) where
according to Eq. (A4), and a rotational disturbance \( { \nabla \times \vec{u} } \) (or a shear wave) may be transmitted with a wave velocity V _{ S } where
according to Eq. (A6). In seismology, and for historical reasons, these two types of waves are called the primary (P) and the secondary (S) waves, respectively.
For a heterogeneous, isotropic, and elastic medium, the equation of motion is more complex than Eq. (A3), and is given by Karal and Keller [65] as
Furthermore, the compressional wave motion is no longer purely longitudinal, and the shear wave motion is no longer purely transverse. A review of seismic wave propagation and imaging in complex media may be found in the entry by Igel et al. Seismic Wave Propagation in Media with Complex Geometries, Simulation of.
A significant portion of seismological research is based on the solution of the elastic wave equations with the appropriate initial and boundary conditions. However, explicit and unique solutions are rare, except for a few simple problems. One approach is to transform the wave equation to the eikonal equation and seek solutions in terms of wave fronts and rays that are valid at high frequencies. Another approach is to develop through specific boundary conditions a solution in terms of normal modes [77]. Although ray theory is only an approximation [17], the classic work of Jeffreys and Bullen, and Gutenberg used it to determine Earth structure and locate earthquakes that occurred in the first half of the 20th century. It remains a principal tool used by seismologists even today. Impressive developments in normal mode and surface wave studies (in both theory and observation) started in the second half of the 20th century, leading to realistic quantification of earthquakes using moment tensor methodology [21].
Rotational Ground Motions
Rotations in ground motion and in structural responses have been deduced indirectly from accelerometer arrays, but such estimates are valid only for long wavelengths compared to the distances between sensors (e. g., [16,34,52,88,90,104]). The rotational components of ground motion have also been estimated theoretically using kinematic source models and linear elastodynamic theory of wave propagation in elastic solids [14,69,70,111].
In the past decade, rotational motions from teleseismic and small local earthquakes were also successfully recorded by sensitive rotational sensors, in Japan, Poland, Germany, New Zealand, and Taiwan (e. g., [53,55,56,105,106,107,108]). The observations in Japan and Taiwan show that the amplitudes of rotations can be one to two orders of magnitude greater than expected from the classical linear theory. Theoretical work has also suggested that, in granular materials or cracked 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 (Earthquake Source: Asymmetry and Rotation Effects).
Because of lack of instrumentation, rotational motions have not yet been recorded in the near‐field (within \( { \sim 25\,\mathrm{km} } \) of fault ruptures) of strong earthquakes (magnitude \( { > 6.5 } \)), where the discrepancy between observations and theoretical predictions may be the largest. Recording such ground motions will require extensive seismic instrumentation along some well‐chosen active faults and luck. To this end, several seismologists have been advocating such measurements, and a current deployment in southwestern Taiwan by its Central Weather Bureau is designed to “capture” a repeat of the 1906 Meishan earthquake (magnitude 7.1) with both translational and rotational instruments.
Rotations in structural response, and the contributions to the response from the rotational components of the ground motion, have also been of interest for many decades (e. g., [78,87,98]. Recent reviews on rotational motions in seismology and on the effects of the rotational components of ground motion on structures can be found, for examples, in Cochard et al. [18] and Pillet and Virieux [93], and Trifunac [112], respectively.
Growing Interest – The IWGoRS
Various factors have led to spontaneous organization within the scientific and engineering communities interested in rotational motions. Such factors include: the growing number of successful direct measurements of rotational ground motions (e. g., by ring laser gyros, fiber optic gyros, and sensors based on electro‐chemical technology); increasing awareness about the usefulness of the information they provide (e. g., in constraining the earthquake rupture properties, extracting information about subsurface properties, and about deformation of structures during seismic and other excitation); and a greater appreciation for the limitations on information that can be extracted from the translational sensors due to their sensitivity to rotational motions e. g., computation of permanent displacements from accelerograms (e. g., [13,39,40,41,93,113]).
A small workshop on Rotational Seismology was organized by W.H.K. Lee, K. Hudnut, and J.R. Evans of the USGS on 16 February 2006 in response to grassroots interest. It was held at the USGS offices in Menlo Park and in Pasadena, California, with about 30 participants from about a dozen institutions participating via teleconferencing and telephone [27]. This event led to the formation of the International Working Group on Rotational Seismology in 2006, inaugurated at a luncheon during the AGU 2006 Fall Meeting in San Francisco.
The International Working Group on Rotational Seismology (IWGoRS) aims to promote investigations of rotational motions and their implications, and the sharing of experience, data, software and results in an open web‐based environment (http://www.rotationalseismology.org). It consists of volunteers and has no official status. H. Igel and W.H.K. Lee currently serve as “co‐organizers”. Its charter is accessible on the IWGoRS web site. The Working Group has a number of active members leading task groups that focus on the organization of workshops and scientific projects, including: testing and verifying rotational sensors, broadband observations with ring laser systems, and developing a field laboratory for rotational motions. The IWGoRS web site also contains the presentations and posters from related meetings, and eventually will provide access to rotational data from many sources.
The IWGoRS organized a special session on Rotational Motions in Seismology, convened by H. Igel, W.H.K. Lee, and M. Todorovska during the 2006 AGU Fall Meeting [76]. The goal of that session was to discuss rotational sensors, observations, modeling, theoretical aspects, and potential applications of rotational ground motions. A total of 21 papers were submitted for this session, and over 100 individuals attended the oral session.
The large attendance at this session reflected common interests in rotational motions from a wide range of geophysical disciplines, including strong‐motion seismology, exploration geophysics, broadband seismology, earthquake engineering, earthquake physics, seismic instrumentation, seismic hazards, geodesy, and astrophysics, thus confirming the timeliness of IWGoRS. It became apparent that to establish an effective international collaboration within the IWGoRS, a larger workshop was needed to allow sufficient time to discuss the many issues of interest, and to draft research plans for rotational seismology and engineering applications.
First International Workshop
The First International Workshop on Rotational Seismology and Engineering Applications was held in Menlo Park, California, on 18–19 September 2007. This workshop was hosted by the US Geological Survey (USGS), which recognized this topic as a new research frontier for enabling a better understanding of the earthquake process and for the reduction of seismic hazards. The technical program consisted of three presentation sessions: plenary (4 papers) and oral (6 papers) held during the first day, and poster (30 papers) held during the morning of the second day. A post‐workshop session was held on the morning of September 20, in which scientists of the Laser Interferometer Gravitational‐wave Observatory (LIGO) presented their work on seismic isolation of their ultra‐high precision facility, which requires very accurate recording of translational and rotational components of ground motions (3 papers). Proceedings of this Workshop were released in Lee et al. [75] with a DVD disc that contains all the presentation files and supplementary information.
One afternoon of the workshop was devoted to in‐depth discussions on the key outstanding issues and future directions. The participants could join one of five panels on the following topics: (1) theoretical studies of rotational motions (chaired by L. Knopoff), (2) measuring far‐field rotational motions (chaired by H. Igel), (3) measuring near‐field rotational motions (chaired by T.L. Teng), (4) engineering applications of rotational motions (chaired by M.D. Trifunac), and (5) instrument design and testing (chaired by J.R. Evans). The panel reports on key issues and unsolved problems, and on research strategies and plans, can be found in Appendices 2.1 through 2.5 in Lee et al. [75]. Following the in‐depth group discussions, the panel chairs reported on the group discussions in a common session, with further discussions among all the participants.
Discussions
Since rotational ground motions may play a significant role in the near‐field of earthquakes, rotational seismology has emerged as a new frontier of research. During the Workshop discussions, L. Knopoff asked: Is there a quadratic rotation‐energy relation, in the spirit of Green's strain‐energy relation, coupled to it or independent of it? Can we write a rotation‐torque formula analogous to Hooke's law for linear elasticity in the form
where ω_{ kl } is the rotation,
L _{ ij } is the torque density; and d _{ ijkl } are the coefficients of rotational elasticity? How are the d's related to the usual c's of elasticity? If we define the rotation vector as
we obtain
where the torque density is \( { \nabla \times \vec{f} } \), \( { \vec{f} } \) is the body force density, and ρ is density of the medium. This shows that rotational waves propagate with S‑wave velocity and that it may be possible to store torques. Eq. (15) is essentially an extension using the classical elasticity theory.
Lakes [67] pointed out that the behavior of solids can be represented by a variety of continuum theories. In particular, the elasticity theory of the Cosserat brothers [19] incorporates (1) a local rotation of points as well as the translation motion assumed in the classical theory, and (2) a couple stress (a torque per unit area) as well as the force stress (force per unit area). In the constitutive equation for the classical elasticity theory, there are two independent elastic constants, whereas for the Cosserat elastic theory there are six. Lakes (personal communication, 2007) advocates that there is substantial potential for using generalized continuum theories in geo‐mechanics, and any theory must have a strong link with experiment (to determine the constants in the constitutive equation) and with physical reality.
Indeed some steps towards better understandings of rotational motions have taken place. For example, Twiss et al. [114] argued that brittle deformation of the Earth's crust (Brittle Tectonics: A Nonlinear Dynamical System) involving block rotations is comparable to the deformation of a granular material, with fault blocks acting like the grains. They realized the inadequacy of classical continuum mechanics and applied the Cosserat or micropolar continuum theory to take into account two separate scales of motions: macro‐motion (large‐scale average motion composed of macrostrain rate and macrospin), and micro‐motion (local motion composed of microspin). A theoretical link is then established between the kinematics of crustal deformation involving block rotations and the effects on the seismic moment tensor and focal mechanism solutions.
Recognizing that rotational seismology is an emerging field, the Bulletin of Seismological Society of America will be publishing in 2009 a special issue under the guest editorship of W.H.K. Lee, M. Çelebi, M.I. Todorovska, and H. Igel.
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Lee, W.H.K., Wu, YM. (2009). Earthquake Monitoring and Early Warning Systems. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/9780387304403_152
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