Tectonic stress and the spectra of seismic shear waves from earthquakes
TL;DR: In this paper, an earthquake model is derived by considering the effective stress available to accelerate the sides of the fault, and the model describes near and far-field displacement-time functions and spectra and includes the effect of fractional stress drop.
Abstract: An earthquake model is derived by considering the effective stress available to accelerate the sides of the fault. The model describes near- and far-field displacement-time functions and spectra and includes the effect of fractional stress drop. It successfully explains the near- and far-field spectra observed for earthquakes and indicates that effective stresses are of the order of 100 bars. For this stress, the estimated upper limit of near-fault particle velocity is 100 cm/sec, and the estimated upper limit for accelerations is approximately 2g at 10 Hz and proportionally lower for lower frequencies. The near field displacement u is approximately given by u(t) = (σ/μ) βr(1 - e−t/r) where. σ is the effective stress, μ is the rigidity, β is the shear wave velocity, and τ is of the order of the dimension of the fault divided by the shear-wave velocity. The corresponding spectrum is
Ω(ω)=σβμ1ω(ω2+τ−2)1/2(1)
The rms average far-field spectrum is given by
〈 Ω(ω) 〉=〈 Rθϕ 〉σβμrRF(e)1ω2+α2(2)
where 〈Rθϕ〉 is the rms average of the radiation pattern; r is the radius of an equivalent circular dislocation surface; R is the distance; F(e) = {[2 – 2e][1 – cos (1.21 eω/α)] +e2}1/2; e is the fraction of stress drop; and α = 2.21 β/r. The rms spectrum falls off as (ω/α)−2 at very high frequencies. For values of ω/α between 1 and 10 the rms spectrum falls off as (ω/α)−1 for e < ∼0.1. At low frequencies the spectrum reduces to the spectrum for a double-couple point source of appropriate moment. Effective stress, stress drop and source dimensions may be estimated by comparing observed seismic spectra with the theoretical spectra.
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25 Jan 1991TL;DR: The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws -producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events.
Abstract: This essential reference for graduate students and researchers provides a unified treatment of earthquakes and faulting as two aspects of brittle tectonics at different timescales. The intimate connection between the two is manifested in their scaling laws and populations, which evolve from fracture growth and interactions between fractures. The connection between faults and the seismicity generated is governed by the rate and state dependent friction laws - producing distinctive seismic styles of faulting and a gamut of earthquake phenomena including aftershocks, afterslip, earthquake triggering, and slow slip events. The third edition of this classic treatise presents a wealth of new topics and new observations. These include slow earthquake phenomena; friction of phyllosilicates, and at high sliding velocities; fault structures; relative roles of strong and seismogenic versus weak and creeping faults; dynamic triggering of earthquakes; oceanic earthquakes; megathrust earthquakes in subduction zones; deep earthquakes; and new observations of earthquake precursory phenomena.
3,802 citations
TL;DR: In this article, an empirical relation involving seismic moment M, energy E, magnitude M, and fault dimension L (or area S) is discussed on the basis of an extensive set of earthquake data (M_S ≧ 6) and simple crack and dynamic dislocation models.
Abstract: Empirical relations involving seismic moment M_o, magnitude M_S, energy E_S and fault dimension L (or area S) are discussed on the basis of an extensive set of earthquake data (M_S ≧ 6) and simple crack and dynamic dislocation models. The relation between log S and log M_o is remarkably linear (slope ∼ 2/3) indicating a constant stress drop Δσ; Δσ = 30, 100 and 60 bars are obtained for inter-plate, intra-plate and “average” earthquakes, respectively. Except for very large earthquakes, the relation M_S ∼ (2/3) log M_o ∼ 2 log L is established by the data. This is consistent with the dynamic dislocation model for point dislocation rise times and rupture times of most earthquakes. For very large earthquakes M_S ∼ (1/3) log M_o ∼ log L ∼ (1/3) log E_S. For very small earthquakes M_S ∼ log M_o ∼ 3 log L ∼ log E_S. Scaling rules are assumed and justified. This model predicts log E_S ∼ 1.5 M_S ∼ 3 log L which is consistent with the Gutenberg-Richter relation. Since the static energy is proportional to σL^3, where σ is the average stress, this relation suggests a constant apparent stress ησ where η is the efficiency. The earthquake data suggest ησ ~ 1/2 Δσ. These relations lead to log S ∼ M_S consistent with the empirical relation. This relation together with a simple geometrical argument explains the magnitude-frequency relation log N ∼ − M_S.
2,648 citations
TL;DR: In this paper, physical factors likely to affect the genesis of the various fault rocks are examined in relation to the energy budget of fault zones, the main velocity modes of faulting and the type of fault, whether thrust, wrench, or normal.
Abstract: Physical factors likely to affect the genesis of the various fault rocks—frictional properties, temperature, effective stress normal to the fault and differential stress—are examined in relation to the energy budget of fault zones, the main velocity modes of faulting and the type of faulting, whether thrust, wrench, or normal. In a conceptual model of a major fault zone cutting crystalline quartzo-feldspathic crust, a zone of elastico-frictional (EF) behaviour generating random-fabric fault rocks (gouge—breccia—cataclasite series—pseudotachylyte) overlies a region where quasi-plastic (QP) processes of rock deformation operate in ductile shear zones with the production of mylonite series rocks possessing strong tectonite fabrics. In some cases, fault rocks developed by transient seismic faulting can be distinguished from those generated by slow aseismic shear. Random-fabric fault rocks may form as a result of seismic faulting within the ductile shear zones from time to time, but tend to be obliterated by continued shearing. Resistance to shear within the fault zone reaches a peak value (greatest for thrusts and least for normal faults) around the EF/QP transition level, which for normal geothermal gradients and an adequate supply of water, occurs at depths of 10–15 km.
1,948 citations
TL;DR: In this article, the authors proposed a frequency-domain scaling model for predicting seismic motions as a function of source strength, which can be applied to any time series having a stochastic character, including ground acceleration, velocity and the oscillator outputs on which response spectra and magnitude are based.
Abstract: Theoretical predictions of seismic motions as a function of source strength are often expressed as frequency-domain scaling models. The observations of interest to strong-motion seismology, however, are usually in the time domain (e.g., various peak motions, including magnitude). The method of simulation presented here makes use of both domains; its essence is to filter a suite of windowed, stochastic time series so that the amplitude spectra are equal, on the average, to the specified spectra. Because of its success in predicting peak and rms accelerations (Hanks and McGuire, 1981), an ω -squared spectrum with a high-frequency cutoff ( f m), in addition to the usual whole-path anelastic attenuation, and with a constant stress parameter (Δ σ ) has been used in the applications of the simulation method. With these assumptions, the model is particularly simple: the scaling with source size depends on only one parameter—seismic moment or, equivalently, moment magnitude. Besides peak acceleration, the model gives a good fit to a number of ground motion amplitude measures derived from previous analyses of hundreds of recordings from earthquakes in western North America, ranging from a moment magnitude of 5.0 to 7.7. These measures of ground motion include peak velocity, Wood-Anderson instrument response, and response spectra. The model also fits peak velocities and peak accelerations for South African earthquakes with moment magnitudes of 0.4 to 2.4 (with f m = 400 Hz and Δ σ = 50 bars, compared to f m = 15 Hz and Δ σ = 100 bars for the western North America data). Remarkably, the model seems to fit all essential aspects of high-frequency ground motions for earthquakes over a very large magnitude range .
Although the simulation method is useful for applications requiring one or more time series, a simpler, less costly method based on various formulas from random vibration theory will often suffice for applications requiring only peak motions. Hanks and McGuire (1981) used such an approach in their prediction of peak acceleration. This paper contains a generalization of their approach; the formulas used depend on the moments (in the statistical sense) of the squared amplitude spectra, and therefore can be applied to any time series having a stochastic character, including ground acceleration, velocity, and the oscillator outputs on which response spectra and magnitude are based .
1,708 citations
TL;DR: In this paper, a plane circular model of a frictional fault using numerical methods was studied and it was shown that the average corner frequency of S waves v s is related to the final source radius, a, by v s = 0.21 β/α.
Abstract: We study a plane circular model of a frictional fault using numerical methods. The model is dynamic since we specify the effective stress at the fault. In one model we assume that the fault appears instantaneously in the medium; in another, that the rupture nucleates at the center and that rupture proceeds at constant subsonic velocity until it suddenly stops. The total source slip is larger at the center and the rise time is also longer at the center of the fault. The dynamic slip overshoots the static slip by 15 to 35 per cent. As a consequence, the stress drop is larger than the effective stress and the apparent stress is less than one half the effective stress.
The far-field radiation is discussed in detail. We distinguish three spectral regions. First, the usual constant low-frequency level. Second, an intermediate region controlled by the fault size and, finally, the high-frequency asymptote. The central region includes the corner frequency and is quite complicated. The corner frequency is shown to be inversely proportional to the width of the far-field displacement pulse which, in turn, is related to the time lag between the stopping phases. The average corner frequency of S waves v s is related to the final source radius, a , by v s = 0.21 β/α . The corner frequency of P waves is larger than v s by an average factor of 1.5.
1,628 citations
References
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TL;DR: In this paper, the authors investigated the dependence of the amplitude spectrum of seismic waves on source size by fitting an exponentially decaying function to the autocorrelation function of the dislocation velocity and found that the most convenient parameter for their purpose is the magnitude Ms, defined for surface waves with period of 20 sec.
Abstract: The dependence of the amplitude spectrum of seismic waves on source size is investigated on the basis of two dislocation models of an earthquake source. One of the models (by N. Haskell) is called the ω³ model, and the other, called the ω² model, is constructed by fitting an exponentially decaying function to the autocorrelation function of the dislocation velocity. The number of source parameters is reduced to one by the assumption of similarity. We found that the most convenient parameter for our purpose is the magnitude Ms, defined for surface waves with period of 20 sec. Spectral density curves are determined for given Ms. Comparison of the theoretical curves with observations is made in two different ways. The observed ratios of the spectra of seismic waves with the same propagation path but from earthquakes of different sizes are compared with the corresponding theoretical ratios, thereby eliminating the effect of propagation on the spectrum. The other method is to check the theory with the empirical relation between different magnitude scales defined for different waves at different periods. The ω² model gives a satisfactory agreement with such observations on the assumption of similarity, but the ω³ model does not. We find, however, some indications of departure from similarity. The efficiency of seismic radiation seems to increase with decreasing magnitude if the Gutenberg-Richter magnitude-energy relation is valid. The assumption of similarity implies a constant stress drop independent of source size. A preliminary study of Love waves from the Parkfield earthquake of June 28, 1966, shows that the stress drop at the source of this earthquake is lower than the normal value (around 100 bars) by about 2 orders of magnitude.
1,352 citations
TL;DR: Stick-slip often accompanies frictional sliding in laboratory experi ments with geologic materials and may represent stick slip during sliding along old or newly formed faults in the earth.
Abstract: Stick-slip often accompanies frictional sliding in laboratory experi ments with geologic materials. Shallow focus earthquakes may represent stick slip during sliding along old or newly formed faults in the earth In such a situation, observed stress drops repre sent release of a small fraction of the stress supported by the rock surround ing the earthquake focus.
868 citations
TL;DR: In this article, the relation of earthquake magnitude M to energy E (in ergs) was investigated and three different magnitude scales were proposed: M_L, the magnitude originally defined by Richter for local earthquakes in California as recorded on standard torsion seismometers, M_S, based on calculated ground amplitudes for surface waves of periods of about 20 sec. in shallow teleseisms, and M_B, that based on the amplitude/period ratio in body waves for both shallow and deep earthquakes.
Abstract: In a paper presented at a meeting of the Seismological Society of America on April 29, 1955, we have revised previous work on the relation of earthquake magnitude M to energy E (in ergs). Methods formerly used to extend the magnitude scale for local earthquakes to teleseisms lead to inconsistencies, so that in effect three different magnitude scales are in use: (1) M_L, the magnitude originally defined by Richter for local earthquakes in California as recorded on standard torsion seismometers. (2) M_S, that based on calculated ground amplitudes for surface waves of periods of about 20 sec. in shallow teleseisms. (3) M_B, that based on the amplitude/period ratio in body waves for both shallow and deep earthquakes.
834 citations
TL;DR: In this paper, it was shown that a shear fault is rigorously equivalent to a distribution of double-couple point sources over the fault plane, while a tensile fault is composed of force dipoles normal to the fault surface with a superimposed purely compressional component.
Abstract: Starting with a Green9s function representation of the solution of the elastic field equations for the case of a prescribed displacement discontinuity on a fault surface, it is shown that a shear fault (relative displacement parallel to the fault plane) is rigorously equivalent to a distribution of double-couple point sources over the fault plane. In the case of a tensile fault (relative displacement normal to the fault plane) the equivalent point source distribution is composed of force dipoles normal to the fault plane with a superimposed purely compressional component. Assuming that the fault break propagates in one direction along the long axis of the fault plane and that the relative displacement at a given point has the form of a ramp time function of finite duration, T , the total radiated P and S wave energies and the total energy spectral densities are evaluated in closed form in terms of the fault plane dimensions, final fault displacement, the time constant T , and the fault propagation velocity. Using fault parameters derived principally from the work of Ben-Menahem and Toksoz on the Kamchatka earthquake of November 4, 1952, the calculated total energy appears to be somewhat low and the calculated energy spectrum appears to be deficient at short periods. It is suggested that these discrepancies are due to over-simplification of the assumed model, and that they may be corrected by (1) assuming a somewhat roughened ramp for the fault displacement time function to correspond to a stick-slip type of motion, and (2) assuming that the short period components of the fault displacement wave are coherent only over distances considerably smaller than the total fault length.
731 citations
TL;DR: In this paper, an explicit expression for the body force to be applied in the absence of a dislocation, which produces radiation identical to that of the dislocation was derived for dislocations in an anisotropic inhomogeneous medium.
Abstract: An explicit expression is derived for the body force to be applied in the absence of a dislocation, which produces radiation identical to that of the dislocation. This equivalent force depends only upon the source and the elastic properties of the medium in the immediate vicinity of the source and not upon the proximity of any reflecting surfaces. The theory is developed for dislocations in an anisotropic inhomogeneous medium; in the examples isotropy is assumed. For displacement dislocation faults, the double couple is an exact equivalent body force.
550 citations