Transformation of multipolar source fields under a change of reference frame

Simple and convenient formulae are derived which describe the transformation of a multipolar expansion under an arbitrary proper rotation of the reference frame. When combined with the corresponding formulae for a translation, these results show how multipolar representations of source fields transform under any proper displacement of the reference frame. Particular emphasis is placed on the seismic source problem; however, these results find applications in many other physical problems

Transient and impulse responses of a one-dimensional linearly attenuating medium — II. A parametric study

We investigate one-dimensional waves in a standard linear solid for geophysically relevant ranges of the parameters. The critical parameters are shown to be T*= t_u/Q_m where t_u is the travel time and Q_m the quality factor in the absorption band, and τ^(-1)_m, the high-frequency cut-off of the relaxation spectrum. The visual onset time, rise time, peak time, and peak amplitude are studied as functions of T* and τ_m. For very small τ_m, this model is shown to be very similar to previously proposed attenuation models. As τ_m grows past a critical value which depends on T*, the character of the attenuated pulse changes. Seismological implications of this model may be inferred by comparing body wave travel times with a ‘one second’ earth model derived from long-period observations and corrected for attenuation effects assuming a frequency independent Q over the seismic band. From such a comparison we speculate that there may be a gap in the relaxation spectrum of the Earth's mantle for relaxation times shorter than about one second. However, observational constraints from the attenuation of body waves suggest that such a gap might in fact occur at higher frequencies. Such a hypothesis would imply a frequency dependence of Q in the Earth's mantle for short-period body waves

Present-day plate motions

A data set comprising 110 spreading rates, 78 transform fault azimuths and 142 earthquake slip vectors was inverted to yield a new instantaneous plate motion model, designated RM2. The mean averaging interval for the relative motion data was reduced to less than 3 My. A detailed comparison of RM2 with angular velocity vectors which best fit the data along individual plate boundaries indicates that RM2 performs close to optimally in most regions, with several notable exceptions. On the other hand, a previous estimate (RM1) failed to satisfy an extensive set of new data collected in the South Atlantic Ocean. It is shown that RM1 incorrectly predicts the plate kinematics in the South Atlantic because the presently available data are inconsistent with the plate geometry assumed in deriving RM1. It is demonstrated that this inconsistency can be remedied by postulating the existence of internal deformation with the Indian plate, although alternate explanations are possible

Transient and impulse responses of a one-dimensional linearly attenuating medium – I. Analytical results

The transient and impulse responses (Green's function) for one-dimensional wave propagation in a standard linear solid are calculated using a Laplace Transform method. The spectrum of relaxation times is chosen so as to model a constant Q medium within an absorption band covering a broad frequency range which may be chosen so as to include the seismic frequencies. The inverse transform may be evaluated asymptotically in the limit of very long propagation times using the saddle point method. For shorter propagation times the method of steepest descent may be modified so as to yield an accurate first motion approximation. The character of the small amplitude precursor to the large amplitude ‘visible’ signal is investigated analytically. It is shown that the signal velocity is intermediate between the high-frequency (‘unrelaxed’) and the low-frequency (‘relaxed’) limits of the phase velocity

Near-field waveforms from an arbitrarily expanding, transparent spherical cavity in a prestressed medium

A simple, approximate (‘transparent’) solution is derived for the near-field radiation emitted by a spherical cavity expanding in an initial pure shear prestress field. Near-field terms, their propagation and decay are discussed for a variety of growth histories, and are shown to be rather insensitive to the detailed variations of rupture velocity. The transparency approximation is shown to be adequate in the near field as well as in the far field; the main effect is a slight narrowing of far-field pulses. Time domain moment estimators at close range are more reliable for the S wave than for the P wave since transverse pulses are not as strongly contaminated by near-field effects

Dynamics in prestressed media with moving phase boundaries: a continuum theory of failure in solids

Spontaneous failure in a solid medium is described as a localized transition of the material from one physical state to another, characterized in part by contrasting rheological properties and density. Such a process is viewed as a local disordering of the relatively ordered structure of the solid due to any variety of causes, such as massive microfracturing or shear melting, and can be confined to a very thin zone, but nevertheless of finite volume such that a volumetric transition energy can be defined. This leads to the description of failure as a generalized phase transition in a prestressed continuum, with instability and transition zone growth being driven by the energy contributions from the relaxation of stress in the surrounding medium. Direct application of mass, momentum and energy conservation to such a generalized phase transition leads to ‘jump’ conditions specified on the growing boundary surface of the transition zone, that relate the rupture growth to discontinuous changes in the dynamic field variables across the failure zone boundary. These field discontinuities are, in turn, related to the localized changes in physical properties induced by failure. Dynamical conditions for rapid spontaneous failure growth in a stressed medium are investigated in some detail, and we find that the failure boundary growth can be simply expressed in terms of energy ‘failure condition’ and a dynamic growth condition specifying the rupture velocity. These results imply that the integral energy change associated with earthquakes is in the range 10^4−10^6 erg/g. Further the failure growth rate is shown to be expressible in terms of the rheological properties of the material before and after failure. For shear melting resulting in a low viscosity fluid, for example, the rupture velocity will be near the shear velocity of the original material. A general Green's function solution for the radiation due to stress relaxation in the medium surrounding the growing failure zone is given and provides the basis for detailed computations of the strain or displacement field changes due to spontaneous failure processes. In particular, it is shown that the jump conditions for the growing transition zone boundary appear naturally as surface integral terms over the boundary. Since these boundary conditions contain the failure rate explicitly, then these terms include effects that have not been represented in previous integral representations of the radiation field resulting from failure. Further, we show that the formal Green's integral representation for the dynamical wave field can be used with known, simple Green's functions to generate approximate solutions for complex failure processes occurring in media with inhomogeneous material properties and prestress

Far-field waveforms from an arbitrarily expanding, transparent spherical cavity in a prestressed medium

Stress relaxation due to a growing cavity in a uniformly prestressed (pure shear) elastic medium is investigated, using the transparent source approximation of Archambeau (1972). A simple representation of the far-field radiation is obtained. A planar, circular dislocation of same growth history as the cavity is constructed which yields the same far-field pulses, except for geometrical effects. It is shown that different ‘equivalent’ dislocations must be used to model P and S pulses wherever the rupture velocity is trans-sonic. Simple analytical forms for the far-field pulses are derived which hold even in the case of relatively complicated source growth history. The dependence of waveforms on the various source parameters is illustrated by selected waveform calculations. It is suggested that this model yields an adequate representation of the far-field tectonic release radiation from underground explosions

Numerical Modelling of Instantaneous Plate Tectonics

Assuming lithospheric plates to be rigid, we systematically invert 68 spreading rates, 62 fracture zones trends and 10^6 earthquake slip vectors simultaneously to obtain a self-consistent model of instantaneous relative motions for eleven major plates. The inverse problem is linearized and solved iteratively by a maximum likelihood procedure. Because the uncertainties in the data are small, Gaussian statistics are shown to be adequate. The use of a linear theory permits (1) the calculation of the uncertainties in the various angular velocity vectors caused by uncertainties in the data, and (2) quantitative examination of the distribution of information within the data set. The existence of a self-consistent model satisfying all the data is strong justification of the rigid plate assumption. Slow movement between North and South America is shown to be resolvable. We then invert the trends of 20 linear island chains and aseismic ridges under the assumptions that they represent the directions of plate motions over a set of hot spots fixed with respect to each other. We conclude that these hot spots have had no significant relative motions in the last 10 My

High-Resolution Topography along Surface Rupture of the 16 October 1999 Hector Mine, California, Earthquake (M_w 7.1) from Airborne Laser Swath Mapping

In order to document surface rupture associated with the Hector Mine earthquake, in particular, the area of maximum slip and the deformed surface of Lavic Lake playa, we acquired high-resolution data using relatively new topographic-mapping methods. We performed a raster-laser scan of the main surface breaks along the entire rupture zone, as well as along an unruptured portion of the Bullion fault. The image of the ground surface produced by this method is highly detailed, comparable to that obtained when geologists make particularly detailed site maps for geomorphic or paleoseismic studies. In this case, however, for the first time after a surface-rupturing earthquake, the detailed mapping is along the entire fault zone rather than being confined to selected sites. These data are geodetically referenced, using the Global Positioning System, thus enabling more accurate mapping of the rupture traces. In addition, digital photographs taken along the same flight lines can be overlaid onto the precise topographic data, improving terrain visualization. We demonstrate the potential of these techniques for measuring fault-slip vectors

Predictability of Self-Organizing Systems

We study the predictability of large events in self-organizing systems. We focus on a set of models which have been studied as analogs of earthquake faults and fault systems, and apply methods based on techniques which are of current interest in seismology. In all cases we find detectable correlations between precursory smaller events and the large events we aim to forecast. We compare predictions based on different patterns of precursory events and find that for all of the models a new precursor based on the spatial distribution of activity outperforms more traditional measures based on temporal variations in the local activity.Comment: 15 pages, plain.tex with special macros included, 4 figure