A new method for analysing nonlinear and non-stationary data has been developed. The key part of the method is the empirical mode decomposition method with which any complicated data set can be dec...

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The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis

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

SUMMARY 
We determine best-fitting Euler vectors, closure-fitting Euler vectors, and a new global model (NUVEL-1) describing the geologically current motion between 12 assumed-rigid plates by inverting plate motion data we have compiled, critically analysed, and tested for self-consistency. We treat Arabia, India and Australia, and North America and South America as distinct plates, but combine Nubia and Somalia into a single African plate because motion between them could not be reliably resolved. The 1122 data from 22 plate boundaries inverted to obtain NUVEL-1 consist of 277 spreading rates, 121 transform fault azimuths, and 724 earthquake slip vectors. We determined all rates over a uniform time interval of 3.0m.y., corresponding to the centre of the anomaly 2A sequence, by comparing synthetic magnetic anomalies with observed profiles. The model fits the data well. Unlike prior global plate motion models, which systematically misfit some spreading rates in the Indian Ocean by 8–12mm yr−1, the systematic misfits by NUVEL-1 nowhere exceed ∼3 mm yr−1. The model differs significantly from prior global plate motion models. For the 30 pairs of plates sharing a common boundary, 29 of 30 P071, and 25 of 30 RM2 Euler vectors lie outside the 99 per cent confidence limits of NUVEL-1. Differences are large in the Indian Ocean where NUVEL-1 plate motion data and plate geometry differ from those used in prior studies and in the Pacific Ocean where NUVEL-1 rates are systematically 5–20 mm yr−1 slower than those of prior models. The strikes of transform faults mapped with GLORIA and Seabeam along the Mid-Atlantic Ridge greatly improve the accuracy of estimates of the direction of plate motion. These data give Euler vectors differing significantly from those of prior studies, show that motion about the Azores triple junction is consistent with plate circuit closure, and better resolve motion between North America and South America. Motion of the Caribbean plate relative to North or South America is about 7 mm yr−1 slower than in prior global models. Trench slip vectors tend to be systematically misfit wherever convergence is oblique, and best-fitting poles determined only from trench slip vectors differ significantly from their corresponding closure-fitting Euler vectors. The direction of slip in trench earthquakes tends to be between the direction of plate motion and the normal to the trench strike. Part of this bias may be due to the neglect of lateral heterogeneities of seismic velocities caused by cold subducting slabs, but the larger part is likely caused by independent motion of fore-arc crust and lithosphere relative to the overriding plate.

Current plate motions

Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems

A simple finite difference approximation to the elastic equations of motion is successfully used to solve various wave-propagation problems possessing analytical solutions. On this basis the method is extended to the problem of Love waves propagating across an ocean-continent type boundary. Numerical propagation of an initial transient solution results in seismograms at various distances. The Fourier transforms of these seismograms are used to calculate phase velocity, group delay, and amplitude-transfer coefficients for various seismometer combinations across the transition region. Some results are' the transition zone can have a small but noticeable effect on phase velocities measured between stations situated away from the zone; a phase-velocity anisotropy exists for waves propagating in opposite directions across the same array. In general, waves propagating into the region of thinning have anomalously high-phase velocities and vice versa. This is especially noticeable for velocities measured over the zone of transition, where the perturbations of the phase velocity of continent-to-ocean waves from the expected local phase velocity can exceed the variations expected from the continent-ocean structural differences. The behavior at a distance from the boundary justifies the well-established method of calculating effective group velocities by means of inverse averages of regional group velocities and also indicates a simple correction for the effect of the ocean-continent boundary that can be applied to calculations of earthquake-source mechanisms based on the method of amplitude equalization.

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Love Waves in Nonuniform Wave Guides: Finite Difference Calculations

Why do stochastic point-source and finite-fault simulation models not agree on the predicted ground motions for moderate earthquakes at large distances? This question was posed by Ken Campbell, who attempted to reproduce the Atkinson and Boore (2006) ground-motion prediction equations for eastern North America using the stochastic point-source program SMSIM (Boore, 2005) in place of the finite- source stochastic program EXSIM (Motazedian and Atkinson, 2005) that was used by Atkinson and Boore (2006) in their model. His comparisons suggested that a higher stress drop is needed in the context of SMSIM to produce an average match, at larger distances, with the model predictions of Atkinson and Boore (2006) based on EXSIM; this is so even for moderate magnitudes, which should be well-represented by a point- source model. Why? The answer to this question is rooted in significant differences between point- source and finite-source stochastic simulation methodologies, specifically as imple- mented in SMSIM (Boore, 2005) and EXSIM (Motazedian and Atkinson, 2005 )t o date. Point-source and finite-fault methodologies differ in general in several important ways: (1) the geometry of the source; (2) the definition and application of duration; and (3) the normalization of finite-source subsource summations. Furthermore, the specific implementation of the methods may differ in their details. The purpose of this article is to provide a brief overview of these differences, their origins, and implica- tions. This sets the stage for a more detailed companion article, "Comparing Stochas- tic Point-Source and Finite-Source Ground-Motion Simulations: SMSIM and EXSIM," in which Boore (2009) provides modifications and improvements in the implementations of both programs that narrow the gap and result in closer agreement. These issues are important because both SMSIM and EXSIM have been widely used in the development of ground-motion prediction equations and in modeling the param- eters that control observed ground motions.

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A guide to differences between stochastic point-source and stochastic finite-fault simulations

We fit an 09 -2 model to response spectra from eight recent Greek earth- quakes ranging in size from M = 5.8 to M = 6.9. The diminution parameter K o was determined for each site, with a value near 0.06 for a typical soil site. The stress parameter (Aa) showed little variation from earthquake to earthquake and had a mean value of 56 bars over all earthquakes. Predictions of peak velocity, peak acceleration, rupture duration, and fault length using the derived stress parameters are consistent with observations. Frequency-dependent site amplifications were included in all es- timates; the combined effect of amplification and attenuation had a maximum value close to a factor of 2.5 for a typical soil site, relative to the motions at the surface of a perfectly elastic uniform half-space composed of materials near the source. The results form the foundation for predictions of strong motions in Greece for distances and magnitudes other than those for which data are available.

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Determination of Δσ and κ0 from response spectra of large earthquakes in Greece

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Prediction of earthquake response spectra

Shear-wave velocities (VS) are widely used for earthquake ground- motion site characterization. VS data are now largely obtained using borehole meth- ods. Drilling holes, however, is expensive. Nonintrusive surface methods are inex- pensive for obtaining VS information, but not many comparisons with direct borehole measurements have been published. Because different assumptions are used in data interpretation of each surface method and public safety is involved in site character- ization for engineering structures, it is important to validate the surface methods by additional comparisons with borehole measurements. We compare results obtained from a particular surface method (array measurement of surface waves associated with microtremor) with results obtained from borehole methods. Using a 10-element nested-triangular array of 100-m aperture, we measured surface-wave phase veloci- ties at two California sites, Garner Valley near Hemet and Hollister Municipal Air- port. The Garner Valley site is located at an ancient lake bed where water-saturated sediment overlies decomposed granite on top of granite bedrock. Our array was deployed at a location where seismic velocities had been determined to a depth of 500 m by borehole methods. At Hollister, where the near-surface sediment consists of clay, sand, and gravel, we determined phase velocities using an array located close to a 60-m deep borehole where downhole velocity logs already exist. Because we want to assess the measurements uncomplicated by uncertainties introduced by the inversion process, we compare our phase-velocity results with the borehole VS depth profile by calculating fundamental-mode Rayleigh-wave phase velocities from an earth model constructed from the borehole data. For wavelengths less than 2 times of the array aperture at Garner Valley, phase-velocity results from array measure- ments agree with the calculated Rayleigh-wave velocities to better than 11%. Mea- surement errors become larger for wavelengths 2 times greater than the array aper- ture. At Hollister, the measured phase velocity at 3.9 Hz (near the upper edge of the microtremor frequency band) is within 20% of the calculated Rayleigh-wave veloc- ity. Because shear-wave velocity is the predominant factor controlling Rayleigh- wave phase velocities, the comparisons suggest that this nonintrusive method can provide VS information adequate for ground-motion estimation.

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Comparison of Phase Velocities from Array Measurements of Rayleigh Waves Associated with Microtremor and Results Calculated from Borehole Shear-Wave Velocity Profiles

David M. Boore

Papers

Love Waves in Nonuniform Wave Guides: Finite Difference Calculations

A guide to differences between stochastic point-source and stochastic finite-fault simulations

Determination of Δσ and κ0 from response spectra of large earthquakes in Greece

Prediction of earthquake response spectra

Comparison of Phase Velocities from Array Measurements of Rayleigh Waves Associated with Microtremor and Results Calculated from Borehole Shear-Wave Velocity Profiles