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

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.

http://www.openseismo.org/contributors/Lee/MoWorking_Backups/Mo2012_0424backup/MoWorking/Papers_NotUsed/0_Theory/928_Bru70_Brune_JGR1970_p4997.pdf

Tectonic stress and the spectra of seismic shear waves from earthquakes

A complete suite of closed analytical expressions is presented for the surface displacements, strains, and tilts due to inclined shear and tensile faults in a half-space for both point and finite rectangular sources. These expressions are particularly compact and free from field singular points which are inherent in the previously stated expressions of certain cases. The expressions derived here represent powerful tools not only for the analysis of static field changes associated with earthquake occurrence but also for the modeling of deformation fields arising from fluid-driven crack sources.

/pdf/surface-deformation-due-to-shear-and-tensile-faults-in-a-2vckanzx6g.pdf

Surface deformation due to shear and tensile faults in a half-space

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

A global plate motion model, named NUVEL-1, which describes current plate motions between 12 rigid plates is described, with special attention given to the method, data, and assumptions used Tectonic implications of the patterns that emerged from the results are discussed It is shown that wide plate boundary zones can form not only within the continental lithosphere but also within the oceanic lithosphere; eg, between the Indian and Australian plates and between the North American and South American plates Results of the model also suggest small but significant diffuse deformation of the oceanic lithosphere, which may be confined to small awkwardly shaped salients of major plates

Current plate motions

Strain accumulation and release at a subduction zone are attributed to stick slip on the main thrust zone and steady aseismic slip on the remainder of the plate interface. This process can be described as a superposition of steady state subduction and a repetitive cycle of slip on the main thrust zone, consisting of steady normal slip at the plate convergence rate plus occasional thrust events that recover the accumulated normal slip. Because steady state subduction does not contribute to the deformation at the free surface, deformation observed there is completely equivalent to that produced by the slip cycle alone. The response to that slip is simply the response of a particular earth model to embedded dislocations. For a purely elastic earth model, the deformation cycle consists of a coseismic offset followed by a linear-in-time recovery to the initial value during the interval between earthquakes. For an elastic-viscoelastic earth model (elastic lithosphere over a viscoelastic asthenosphere), the postearthquake recovery is not linear in time. Records of local uplift as a function of time indicate that the long-term postseimic recovery is approximately linear, suggesting that elastic earth models are adequate to describe the deformation cycle. However, the deformation predicted for amore » simple elastic half-space earth does not reproduce the deformation observed along the subduction zones in Japna at all well if stick slip is restricted to the main thrust zone. As recognized earlier by Shimazaki, Seno, and Kato, the uplift profiles could be explained if stick slips were postulated to extend along the plate interface beyond the main thrust zone to a depth of perhaps 100 km, but independent evidence suggests that stick slip at such depths if unlikely.« less

A dislocation model of strain accumulation and release at a subduction zone

Geodetic data along the San Andreas fault between Parkfield and San Francisco, California (latitudes 36°N and 38°N, respectively), have been re-examined to estimate the current relative movement between the American and Pacific plates across the San Andreas fault system. The average relative right lateral motion is estimated to be 32 ± 5 mm/yr for the period 1907–1971. Between 36°N and 37°N it appears that most, if not all, of the plate motion is accommodated by fault creep. Although strain is presumably accumulating north of 37°N (San Francisco Bay area), the geodetic evidence for accumulation is not conclusive.

/pdf/geodetic-determination-of-relative-plate-motion-in-central-3dlm846alw.pdf

Geodetic determination of relative plate motion in central California

A simple two-dimensional model of the earthquake cycle (preearthquake strain accumulation, coseismic strain release, and postseismic readjustment) has been constructed from the Nur-Mavko solution for a screw dislocation in an elastic plate (lithosphere) overlying a viscoelastic substrate (asthenosphere). The deformation at the free surface is calculated for an earthquake cycle imposed by prescribed slip on a transform fault. This deformation is compared to that produced by a similar cycle in an elastic half space so that the effects of viscoelastic relaxation in the asthenosphere may be isolated. The following conclusions are drawn: (1) The surface deformation produced by viscoelastic relaxation in the asthenosphere can be duplicated identically by a reasonable distribution of slip at depth on a vertical fault in an elastic half space. Thus differentiation of two possible modes of postearthquake readjustment will be difficult. (2) The effect of asthenosphere relaxation is important only if the depth of the seismic zone is comparable to the thickness of the lithosphere. If the seismic zone is 15 km deep and the lithosphere is 75 km thick, as commonly estimated for the San Andreas fault zone, asthenosphere relaxation is not particularly significant in determining surface deformation. (3) In a periodic sequence of earthquakes the principal observable effects of viscoelasticity in the asthenosphere are to produce a rapid postearthquake deformation and to concentrate strain accumulation and relaxation even closer to the fault than in the elastic half-space model.

Asthenosphere readjustment and the earthquake cycle

The Chilean earthquake sequence of May 21–22, 1960, was accompanied by linear zones of tectonic warping, including both uplift and subsidence relative to sea level. The region involved is more than 200 km wide and about 1000 km long, and lies along the continental margin between latitude 37° and 48° S. Significant horizontal strains accompanied the vertical movements in parts of the subsided zone for which triangulation data are available. Displacements were initiated near the northern end of the deformed region during the opening earthquake of the sequence (M s ≅ 7.5) on May 21 at 10h 02m 50s GMT and were extended over the remainder of the region during the culminating shock (M s ≅ 8.5) on May 22 at 19h llm 17s GMT. During the latter event, sudden uplift of adjacent portions of the continental shelf and much or all of the continental slope apparently generated the destructive tsunami that immediately followed the main shock. Available data suggest that the primary fault or zone of faulting along which displacement occurred probably is a complex thrust fault roughly 1000 km long and at least 60 km wide; it dips eastward at a moderate angle beneath the continental margin and intersects the surface on the continental slope. Dip slip required to satisfy the surface displacements is at least 20 m and perhaps as large as 40 m. There is some evidence that there was a minor component of right-lateral slip on the fault plane.

Mechanism of the Chilean Earthquakes of May 21 and 22, 1960

A fault surface may be represented by a rectangular surface of horizontal length 2L, width W, and dip δ embedded in an elastic half-space with the top of the fault a depth h below the free surface. The vertical displacement of the free surface for a dip-slip motion Δu on such a fault surface can be calculated from the theory of Maruyama. This calculation has been made for fault models of three different earthquakes, and the results compared with the observed surface deformation in each case. For each calculation the dip of the fault plane was taken from the P wave fault plane solution. The preferred fault models are as follows: Alaska earthquake of March 28, 1964 (magnitude 8.4), 2L = 600 km, W = 200 km, δ = 9°, h = 20 km, and Δu = 10 m. Fairview Peak, Nevada, earthquake of December 16, 1954 (magnitude 7.1), 2L = 24 to 48 km, W = 6 km, δ = 62°, h = 0 km, and Δu = 2 m. Hebgen Lake, Montana, earthquake of August 18, 1959 (magnitude 7.1), 2L = 30 km, W = 15 km, δ = 54°, h = 0.4 km, and Δu = 10 m.

James C. Savage

Papers

A dislocation model of strain accumulation and release at a subduction zone

Geodetic determination of relative plate motion in central California

Asthenosphere readjustment and the earthquake cycle

Mechanism of the Chilean Earthquakes of May 21 and 22, 1960

Surface deformation associated with dip‐slip faulting