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

The nearly coincident forms of the relations between seismic moment M_0 and the magnitudes M_L, M_S, and M_w imply a moment magnitude scale M = ⅔ log M_0 - 10.7 which is uniformly valid for 3 ≲ M_L ≲ 7, 5 ≲ M_s ≲ 7½, and M_w ≳ 7½.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/JB084iB05p02348%4010.1002/%28ISSN%292169-9356.FAULT1

A moment magnitude scale

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.

/pdf/theoretical-basis-of-some-empirical-relations-in-seismology-1e9arnhgn6.pdf

Theoretical basis of some empirical relations in seismology

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

The two-dimensional scattering and diffraction of plane SH waves by a semi-cylindrical canyon is analysed for a general angle of wave incidence The closed-form solution of the problem shows that the surface topography can have prominent effects on incident waves only when the wavelengths of incident motion are short compared to the radius of a canyon The surface amplification of displacement amplitudes around and in the canyon changes rapidly from one point to another, but the amplification is always less than 2 The over-all trends of amplification pattern are determined by two principal parameters: (1) γ, the angle of incidence of plane SH waves, and (2) η, the ratio of radius of the canyon to one-half wave length of incident waves The higher η leads to greater complexity of the pattern of surface displacement amplitudes characterized by more abrupt changes of amplification from one point to another, while γ mainly determines the over-all trends of displacement amplitudes For grazing and nearly grazing incidences, for example, a strong shadow zone is developed behind the canyon



The qualitative analysis of the topographic effects on the Pacoima Dam accelerogram,1 based on the semi-cylindrical canyon, suggests that this strong-motion record was not seriously affected by surface topography of the recording site

Scattering of plane sh waves by a semi‐cylindrical canyon

Correlations of the recorded peak acceleration, velocity and displacement, and the Modified Mercalli intensity have been carried out for 57 earthquakes and 187 strong-motion accelerograms recorded in the Western United States. Correlations of peak acceleration with intensity, characterized by the data scatter exceeding one order of magnitude, have lead to average peak accelerations which are higher than those reported by a majority of previous investigators. New correlations, also characterized by scatter of data of about one order of magnitude, have been presented for peak velocities and displacements of strong ground motion versus Modified Mercalli intensity.

Grouping of all recorded data according to the geology underlying the strong-motion accelerograph stations was carried out and permitted a study of the possible effects that local geology might have on the peaks of strong-motion acceleration, velocity, and displacement. Results of this analysis are as follows: (1) For ground shaking of a particular Modified Mercalli intensity, average peak acceleration recorded on hard rock is higher by a factor less than about two than the average peak acceleration recorded on alluvium; (2) the effect of local geology on the average peak velocity leads to marginally higher peak values on alluvium; and (3) the peak ground displacements are larger, by a factor less than two, when recorded on alluvium rather than on hard rock.

On the correlation of seismic intensity scales with the peaks of recorded strong ground motion

By using the exact analytical solution for the two-dimensional SH-wave propagation in and around an elastic inclusion whose cross section corresponds to one half of an ellipse, we have examined those aspects of the resulting ground motion that are of special interest for earthquake engineering and strong-motion seismology. Computed amplitudes and phases of periodic ground motion display complicated wave-interference phenomena that lead to nearly-standing wave patterns, abrupt changes in the amplification of incident motions along the free surface of the alluvial valley and strong dependence of the overall motions on the incidence angle of SH waves. By comparing the amplification patterns derived from the exact model with the amplifications computed on the basis of an equivalent single-layer model excited by the vertically incident shear waves, we have demonstrated that this approximate representation may lead to meaningful results only if the wavelength of incident waves is longer than the characteristic dimension of the alluvial valley. Although simple, we expect that the model presented in this paper might explain qualitatively the vibrations of some alluvial valleys excited by SH components of strong ground motion.

Surface motion of a semi-elliptical alluvial valley for incident plane SH waves

Scattering of plane sh waves by a semi‐elliptical canyon

A new method is proposed for standard baseline correction of strong-motion accelerograms. It is based on high-pass filtering of the uncorrected digitized accelerogram data. Unlike the parabolic baseline correction, the new method has well-defined frequency transfer function properties which are largely independent of the record length.

/pdf/zero-baseline-correction-of-strong-motion-accelerograms-1aordzf9ef.pdf

Mihailo D. Trifunac

Papers

Scattering of plane sh waves by a semi‐cylindrical canyon

On the correlation of seismic intensity scales with the peaks of recorded strong ground motion

Surface motion of a semi-elliptical alluvial valley for incident plane SH waves

Scattering of plane sh waves by a semi‐elliptical canyon

Zero baseline correction of strong-motion accelerograms