Source parameters for historical earthquakes worldwide are compiled to develop a series of empirical relationships among moment magnitude ( M ), surface rupture length, subsurface rupture length, downdip rupture width, rupture area, and maximum and average displacement per event. The resulting data base is a significant update of previous compilations and includes the additional source parameters of seismic moment, moment magnitude, subsurface rupture length, downdip rupture width, and average surface displacement. Each source parameter is classified as reliable or unreliable, based on our evaluation of the accuracy of individual values. Only the reliable source parameters are used in the final analyses. In comparing source parameters, we note the following trends: (1) Generally, the length of rupture at the surface is equal to 75% of the subsurface rupture length; however, the ratio of surface rupture length to subsurface rupture length increases with magnitude; (2) the average surface displacement per event is about one-half the maximum surface displacement per event; and (3) the average subsurface displacement on the fault plane is less than the maximum surface displacement but more than the average surface displacement. Thus, for most earthquakes in this data base, slip on the fault plane at seismogenic depths is manifested by similar displacements at the surface. Log-linear regressions between earthquake magnitude and surface rupture length, subsurface rupture length, and rupture area are especially well correlated, showing standard deviations of 0.25 to 0.35 magnitude units. Most relationships are not statistically different (at a 95% significance level) as a function of the style of faulting: thus, we consider the regressions for all slip types to be appropriate for most applications. Regressions between magnitude and displacement, magnitude and rupture width, and between displacement and rupture length are less well correlated and have larger standard deviation than regressions between magnitude and length or area. The large number of data points in most of these regressions and their statistical stability suggest that they are unlikely to change significantly in response to additional data. Separating the data according to extensional and compressional tectonic environments neither provides statistically different results nor improves the statistical significance of the regressions. Regressions for cases in which earthquake magnitude is either the independent or the dependent parameter can be used to estimate maximum earthquake magnitudes both for surface faults and for subsurface seismic sources such as blind faults, and to estimate the expected surface displacement along a fault for a given size earthquake.

New Empirical Relationships among Magnitude, Rupture Length, Rupture Width, Rupture Area, and Surface Displacement

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.

The Mechanics of Earthquakes and Faulting

This paper reviews rock friction and the frictional properties of earthquake faults. The basis for rate- and state-dependent friction laws is reviewed. The friction state variable is discussed, including its interpretation as a measure of average asperity contact time and porosity within granular fault gouge. Data are summarized showing that friction evolves even during truly stationary contact, and the connection between modern friction laws and the concept of “static” friction is discussed. Measurements of frictional healing, as evidenced by increasing static friction during quasistationary contact, are reviewed, as are their implications for fault healing. Shear localization in fault gouge is discussed, and the relationship between microstructures and friction is reviewed. These data indicate differences in the behavior of bare rock surfaces as compared to shear within granular fault gouge that can be attributed to dilation within fault gouge. Physical models for the characteristic friction distance are discussed and related to the problem of scaling this parameter to seismic faults. Earthquake afterslip, its relation to laboratory friction data, and the inverse correlation between afterslip and shallow coseismic slip are discussed in the context of a model for afterslip. Recent observations of the absence of afterslip are predicted by the model.

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Laboratory-derived friction laws and their application to seismic faulting

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 .

Stochastic simulation of high-frequency ground motions based on seismological models of the radiated spectra

This paper contains ground-motion prediction equations (GMPEs) for average horizontal-component ground motions as a function of earthquake magnitude, distance from source to site, local average shear-wave velocity, and fault type. Our equations are for peak ground acceleration (PGA), peak ground velocity (PGV), and 5%-damped pseudo-absolute-acceleration spectra (PSA) at periods between 0.01 s and 10 s. They were derived by empirical regression of an extensive strong-motion database compiled by the “PEER NGA” (Pacific Earthquake Engineering Research Center’s Next Generation Attenuation) project. For periods less than 1s , the analysis used 1,574 records from 58 mainshocks in the distance range from 0 km to 400 km (the number of available data decreased as period increased). The primary predictor variables are moment magnitude M, closest horizontal distance to the surface projection of the fault plane R JB , and the time-averaged shear-wave velocity from the surface to 30 m VS30. The equations are applicable for M =5–8 , RJB 200 km, and VS30= 180– 1300 m / s. DOI: 10.1193/1.2830434

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Ground-motion prediction equations for the average horizontal component of PGA, PGV, and 5%-damped PSA at spectral periods between 0.01 s and 10.0 s

The far-field body wave radiation from a class of circular rupture models is investigated as a function of takeoff angle, rupture velocity, and stopping behavior. In particular, the variation of spectral shape, pulse shape, and energy flux over the focal sphere is quantified. These results provide two new methods for estimating the source dimension, the first through the inversion of a characteristic frequency, and the second using the rise time of the displacement pulse shape. The class of kinematic rupture models which has been constructed also allows a direct estimate of the dynamic stress drop using the velocity pulse shape: as an example, this technique is applied to an SMA1 recording of a high stress drop Aleutian earthquake, Finally, the introduction of a new spectral parameter, the integral of the square of the ground velocity, is shown to considerably simplify calculation of the total radiated energy. Together with a re-evaluation of corner frequency and spectral falloff, these new results constitute a spectral theory for circular seismic sources which includes directivity and is valid for a range of subsonic rupture velocities.

A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy

Averages of P - and S -wave radiation patterns over all azimuths and various ranges of takeoff angles (corresponding to observations at teleseismic, regional, and near distances) have been computed for use in seismological applications requiring average radiation coefficients. Various fault orientations and averages of the squared, absolute, and logarithmic radiation patterns have been considered. Effective radiation patterns combining high-frequency direct and surfacere-flected waves from shallow faults have also been derived and used in the computation of average radiation coefficients at teleseismic distances. 

In most cases, the radiation coefficients are within a factor of 1.6 of the commonly used values of 0.52 and 0.63 for the rms of P - and S -wave radiation patterns, respectively, averaged over the whole focal sphere. The main exceptions to this conclusion are the coefficients for P waves at teleseismic distances from vertical strike-slip faults, which are at least a factor of 2.8 smaller than the commonly used value.

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Average body-wave radiation coefficients

Fractal theory is applied in a quantitative analysis of the San Andreas fault (SAF) geometry. The method, which directly measures the increase in total fault length with a decrease in ruler size, gives the fractal dimension D and scaling properties for the chosen length band 0.5–1000 km. A physical interpretation of D is that it measures the irregularity of the fault trace in the selected band. The fault is subdivided into six segments of distinctive seismic behavior. A “main” fault trace which shows either maximum coseismic slip or creeping was selected for analysis, with three alternative branches examined for the SAF system south of San Bernardino. Branches of the fault trace are not considered in this work. Fractal dimensions calculated for the different segments range from 1.0008 to 1.0191, and are different from 1.0 to the 95% confidence interval. These small changes reflect the overall smoothness of the main fault trace. Significant variations in D among segments indicate heterogeneities in the fault smoothness along strike. D also changes significantly from the short-length band to the long-length band where the demarcation point ranges from 1 to 2 km. The short-length band has larger D values. A slight correlation is obtained between the fractal dimension of the main trace and the extent of subparallel faulting. This indicates some correspondence between the main fault trace irregularity and the complexity of subsidiary fault traces in plan view.

Fractal analysis applied to characteristic segments of the San Andreas Fault

We use a database of more than 80 finite-source rupture models for more than 50 earthquakes ( M w 4.1–8.1) with different faulting styles occurring in both tectonic and subduction environments to analyze the location of the hypocenter within the fault and to consider the correlation between hypocenter location and regions of large slip. Rupture in strike-slip and crustal dip-slip earthquakes tends to nucleate in the deeper sections of the fault; subduction earthquakes do not show this tendency. Ratios of the hypocentral slip to either the average or the maximum slip show that rupture can nucleate at locations with any level of relative displacement. Rupture nucleates in regions of very large slip ( D ≥ 2/3 D max ) in only 16% of the events, in regions of large slip (1/3 D max D D max ) in 35% of the events, and in regions of low slip ( D ≤ 1/3 D max ) in 48% of the events. These percentages significantly exceed the percentages of fault area with very large (∼7%) and large (∼28%) slip. Ruptures that nucleate in regions of low slip, however, tend to nucleate close to regions of large slip and encounter a zone of very large slip within half the total rupture length. Applying several statistical tests we conclude that hypocenters are not randomly located on a fault but are located either within or close to regions of large slip.

Hypocenter locations in finite-source rupture models

A statistically complete methodology for the analysis of SMA-1 data, including a direct inversion of the amplitude spectra, is developed and applied to the records of two small earthquakes ( M L = 1.5 in Attica and M L = 2.2 in Blue Mountain Lake) obtained at epicentral distances of ≈1 km. Using ray dependent corner frequency-radius relations, the fault radii are determined to be a = 80 ± 15 m and 25 ± 10 m. The stress drops vary by an order of magnitude; Δσ = 6 ± 4 bars and 85 ± 60 bars. Using hydrofracture data to estimate σ ¯ , the average stress, the seismic efficiency of the Attica event is determined to be ≈0.08. Because of a high estimate of the fracture energy associated with its rupture, the Blue Mountain Lake event is interpreted as having fractured fresh rock. The stopping character of the two events, inferred from their spectral shapes, is shown to differ considerably. These differences are used to contrast the stress and frictional regimes of the two source regions.

John Boatwright

Papers

A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy

Average body-wave radiation coefficients

Fractal analysis applied to characteristic segments of the San Andreas Fault

Hypocenter locations in finite-source rupture models

Detailed spectral analysis of two small New York state earthquakes