Leon Knopoff

Observation and inversion of surface-wave dispersion

Abstract Rayleigh wave phase velocities have been measured to periods in excess of 160 sec for a number of profiles, worldwide and for many different geologic structures. It is found that at least five structural provinces must be established; these include shields, aseismic continental platforms, rifts, ocean basins, and mountains. The inversion of these profiles shows that the shields probably have a little or no low-velocity channel in the mantle, the aseismic platforms have a low-velocity channel with high contrast to a thick lid, the oceans have a high-contrast channel and a thin lid, the rifts have low-velocity material but probably without a lid. Comparison is made between regional observations and the great-circle decomposition methods for determining phase velocities in type regions.

Rank‐ordering statistics of extreme events: Application to the distribution of large earthquakes

Rank-ordering statistics provide 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 worldwide (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 b2 = 2.3 ± 0.3 for large shallow earthquakes and b1 = 1.00 ± 0.02 for smaller shallow earthquakes. However, the crossover magnitude between the two distributions is ill defined. The data available at present do not provide a strong constraint on the crossover 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 worldwide 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 two branches or that a rolloff in the distribution is needed; for this catalog, b = 1.00 ± 0.02 up to the largest magnitude observed, MW ≃ 7.5; hence we conclude that the thickness of the seismogenic layer has no observable influence whatsoever on the frequency distribution in this region.

The acoustic component of western consonance

Abstract A formalism is developed for providing a measure of dissonance in a superposition of complex tones. The formalism is based on an extension of the Helmholtz‐Plomp and Levelt model of beating as the cause of dissonance. For dyads this measure of dissonance gives a good fit to psychological rank orderings of dissonance and its absence (consonance), and to orderings of consonance and dissonance found in Western common practice and pedagogy. A logarithmic scale for the perception of consonance and dissonance is indicated.

Transmission and reflection of Rayleigh waves by wedges

Experimental observations have been made of the transmission and reflection of Rayleigh waves by wedges. Results are reported for Rayleigh waves in aluminum wedges. It is observed that the wave shapes of the transmitted and reflected waves differ from that of the incident wave and depend on the angle of the wedge. The change of shape is attributed to an interference between part of the incident wave‐form and the radiation from a line source placed at the vertex. A procedure is given for the calculation of the partition between the two terms.

The equivalent strength of geometrical barriers to earthquakes

We present a quantitative framework for evaluating the influence of non-planar fault geometry on repeated seismic ruptures. We model quasistatic ruptures on a non-planar fault trace imbedded in a two-dimensional elastic medium under in-plane strain. Because of the presence of fault segments that are not parallel to the regional shear stress (i.e. bends), the apparent strength at a given point on the fault is not fixed, but fluctuates with normal stress. Compressional features behave as increasingly strong barriers to fracture unless the stored normal stress is released in order to unlock the fault. Since slip on the fault itself cannot get rid of the normal stress, this is achieved through the action of off-fault morphological features such as secondary faulting, folding and vertical motions, that we introduce parametrically in the form of an aseismic relaxation. The apparent strength of a fault bend will stabilize in a narrow interval of values after repeated ruptures, characterized by a non-dimensional “hardness” parameter, whereby the relaxation rate is scaled by the tectonic loading rate. On a fault structure having several small, widely separated bends, three families of events can be identified whose frequency and magnitude depend on the hardness (relaxation) parameter and the geometry: small events that cluster in the tension zones of the bends, intermediate size ruptures involving a single interbend segment, and large ruptures that break through bends and link on or more interbend segments. Large multi-segment events are most likely to occur for low values of the hardness, i.e., fast relaxation and slow loading rate. Regions with compressional features act as barriers that stop most ruptures; stress is stored at these sites until they themselves break and initiate motion on the smoother, long reaches of the fault.

SCATTERING OF COMPRESSION WAVES BY SPHERICAL OBSTACLES

The scattering of plane P waves by a spherical obstacle is formulated. A calculation is given for the special case of scattering by a perfectly rigid sphere in which the medium outside has a Poisson’s ratio of 14. The range of sizes of obstacles used in the calculation includes radii very small compared with wave length and radii comparable to the wave length. For incident P waves, scattered P and S are computed with shifts in time phase occurring in both with respect to the incident beam. For small obstacles, the scattered S wave is generally broadside to the scattered P‐wave beam.

Scattering of Impulsive Elastic Waves by a Rigid Cylinder

The exact solution to the problem of the scattering of compressional elastic waves from a line source by a rigid, infinitely dense cylinder imbedded in an isotropic, homogeneous, perfectly elastic medium is obtained in integral form. The integrals are evaluated asymptotically obtain the motions on the wave fronts. In the illuminated zone the saddle point method of integration yields the geometrical optics approximation to the reflected field. In the shadow zone the diffracted field is obtained by evaluating the integrals by the method of Dougall and Watson. In the case of an incident P wave the observed events in the illuminated zone are (1) direct P, (2) reflected P, and (3) reflected S. In the shadow zone the observed events are (1) diffracted P and (2) diffracted S. Both diffracted wave fronts travel around the cylinder with the velocity of P waves.

SCATTERING OF SHEAR WAVES BY SPHERICAL OBSTACLES

The problem of the scattering of plane S waves by a perfectly rigid, infinitely dense sphere is formulated. Calculations are made for the case in which the medium outside the sphere has a Poisson’s ratio of 14. The range of sizes of obstacles used in the calculations includes radii very small compared with the wave length and radii comparable to the wave length. The scattered wave motions include a P mode and two S modes. One of the S modes has a formal correspondence to the SH mode of plane seismology; the other corresponds to the SV mode. At large distances from the obstacle the scattered P and S fields are computed together with the phase shifts in time occurring in all the components. For small obstacles, the scattered azimuthal S component is circularly symmetric; the scattered meridional S component diffraction pattern is generally elongated in the direction of propagation; the scattered P component is generally broadside to the direction of propagation.

The elastic modulus of media containing strongly interacting antiplane cracks

We calculate the elastic modulus for up to 10000 randomly oriented, strongly interacting, nonintersecting, antiplane cracks that have a logarithmic size distribution for a range of concentrations c from 0 to 2. An antiplane boundary integral method is used to compute elemental dislocations on the crack faces. The ratio of slips in the cases of interacting cracks to those for isolated cracks has a nearly unit average. The effective modulus is well fit by a mean-field model in which the cracks do not interact. Prom Gausss theorem in two dimensions, we can show that the mean-field approximation is appropriate for the problem of the modulus of a high concentration of randomly distributed cracks. The mean total field is the external stress field. The modulus as a function of concentration is then simply = μ0/(1 + c/2). This expression differs from the self-consistent result in that the modulus does not become zero at finite concentration. It also differs from modifications to the self-consistent method which predict an exponential decay of modulus with c.

First Motion Methods in Theoretical Seismology

Techniques are presented for approximating integral solutions to some problems in theoretical seismology. The approximations obtained are the first terms of asymptotic series in powers of t−t0, where t is the time and t0 is an arrival time. The approximations are obtained by evaluating the integral form of the Laplace transform of the time solution by the saddle point method or a variation of it. To the resulting expression is applied a Tauberian limit theorem from which is obtained the time solution. Two examples are given which illustrate some of the specific techniques for the use of the method.