A matrix method for elastic wave problems

The solution to problems of elastic wave propagation in multilayered media, in which each layer is homogeneous and where the ensemble of layers has physical properties that vary only with one coordinate, may be given as the quotient of products of matrices. In the case of SH waves, the matrices are of order two; in the case of P-SV waves the matrices are of order four. The individual matrix elements are themselves determinants of order two or four in the two cases. The solution is obtained by means of Laplace's development by minors

Observations of multiple seismic events

Two or more dispersed wave trains each with constant amplitude will interfere giving a resultant wave train which is amplitude modulated, if the individual waves have their principal energies in a common frequency band and if the trains arrive with time separations small compared to their total length. The dispersive characteristics of the trains need not be the same. If the component trains are of comparable magnitude, the modulation due to interference becomes significant and a "beat" phenomenon occurs. Multiple trains of dispersed seismic surface waves may occur because of a temporal and/or spatial distribution at the source or because of multipath propagation. Each of these causal mechanisms influences the amplitude and phase spectra of the resultant wave train; derived properties such as phase velocities and amplitude ratios are also influenced. In the case of multipath propagation, wavelength dependent time delays may occur. Two cases of twin earthquakes are analyzed, and the significant features of interference are demonstrated. In one case, estimates are obtained for the amplitude ratio and time delay of the second shock with respect to the first. The interpretation of seismograms and spectra influenced by multiple events is discussed

First motions from seismic sources near a free surface

The radiation patterns of first motions are calculated for the sudden occurrence of an arbitrarily oriented fault (dislocation) at the surface of a half space; the dislocation in the fault plane is also arbitrarily oriented and is assumed to occur over a very small area of the fault plane. Initially the source is considered at a finite depth and the solution is obtained by allowing the depth to tend to zero. In general the results show a surprising directionality for the radiation of SV. In the focal plane projection the first motions of P and SH for a strike-slip fault show the familiar four-lobed radiation patterns. The first motions of SV show some reversals in polarity with angular distance from the source. The first motions for all components of motion for a dip-slip fault have characteristics governed strongly by the presence of the free surface, and hence differ markedly from the usual radiation patterns for a deeply imbedded source

Rank-Ordering Statistics of Extreme Events: Application to the Distribution of Large Earthquakes

Rank-ordering statistics provides a perspective on the rare, largest elements of a population, whereas the statistics of cumulative distributions are dominated by the more numerous small events. The exponent of a power law distribution can be determined with good accuracy by rank-ordering statistics from the observation of only a few tens of the largest events. Using analytical results and synthetic tests, we quantify the systematic and the random errors. We also study the case of a distribution defined by two branches, each having a power law distribution, one defined for the largest events and the other for smaller events, with application to the World-Wide (Harvard) and Southern California earthquake catalogs. In the case of the Harvard moment catalog, we make more precise earlier claims of the existence of a transition of the earthquake magnitude distribution between small and large earthquakes; the $b$-values are $b_2 = 2.3 \pm 0.3$ for large shallow earthquakes and $b_1 = 1.00 \pm 0.02$ for smaller shallow earthquakes. However, the cross-over magnitude between the two distributions is ill-defined. The data available at present do not provide a strong constraint on the cross-over which has a $50\%$ probability of being between magnitudes $7.1$ and $7.6$ for shallow earthquakes; this interval may be too conservatively estimated. Thus, any influence of a universal geometry of rupture on the distribution of earthquakes world-wide is ill-defined at best. We caution that there is no direct evidence to confirm the hypothesis that the large-moment branch is indeed a power law. In fact, a gamma distribution fits the entire suite of earthquake moments from the smallest to the largest satisfactorily. There is no evidence that the earthquakes of the Southern California catalog have a distribution with tw

Scale free networks of earthquakes and aftershocks

We propose a new metric to quantify the correlation between any two earthquakes. The metric consists of a product involving the time interval and spatial distance between two events, as well as the magnitude of the first one. According to this metric, events typically are strongly correlated to only one or a few preceding ones. Thus a classification of events as foreshocks, main shocks or aftershocks emerges automatically without imposing predefined space-time windows. To construct a network, each earthquake receives an incoming link from its most correlated predecessor. The number of aftershocks for any event, identified by its outgoing links, is found to be scale free with exponent $\gamma = 2.0(1)$. The original Omori law with $p=1$ emerges as a robust feature of seismicity, holding up to years even for aftershock sequences initiated by intermediate magnitude events. The measured fat-tailed distribution of distances between earthquakes and their aftershocks suggests that aftershock collection with fixed space windows is not appropriate.Comment: 7 pages and 7 figures. Submitte

Persistence and Quiescence of Seismicity on Fault Systems

We study the statistics of simulated earthquakes in a quasistatic model of two parallel heterogeneous faults within a slowly driven elastic tectonic plate. The probability that one fault remains dormant while the other is active for a time Dt following the previous activity shift is proportional to the inverse of Dt to the power 1+x, a result that is robust in the presence of annealed noise and strength weakening. A mean field theory accounts for the observed dependence of the persistence exponent x as a function of heterogeneity and distance between faults. These results continue to hold if the number of competing faults is increased. This is related to the persistence phenomenon discovered in a large variety of systems, which specifies how long a relaxing dynamical system remains in a neighborhood of its initial configuration. Our persistence exponent is found to vary as a function of heterogeneity and distance between faults, thus defining a novel universality class.Comment: 4 pages, 3 figures, Revte