Showing papers in "Bulletin of the Seismological Society of America in 1964"

Total energy and energy spectral density of elastic wave radiation from propagating faults

Abstract: Starting with a Green9s function representation of the solution of the elastic field equations for the case of a prescribed displacement discontinuity on a fault surface, it is shown that a shear fault (relative displacement parallel to the fault plane) is rigorously equivalent to a distribution of double-couple point sources over the fault plane. In the case of a tensile fault (relative displacement normal to the fault plane) the equivalent point source distribution is composed of force dipoles normal to the fault plane with a superimposed purely compressional component. Assuming that the fault break propagates in one direction along the long axis of the fault plane and that the relative displacement at a given point has the form of a ramp time function of finite duration, T , the total radiated P and S wave energies and the total energy spectral densities are evaluated in closed form in terms of the fault plane dimensions, final fault displacement, the time constant T , and the fault propagation velocity. Using fault parameters derived principally from the work of Ben-Menahem and Toksoz on the Kamchatka earthquake of November 4, 1952, the calculated total energy appears to be somewhat low and the calculated energy spectrum appears to be deficient at short periods. It is suggested that these discrepancies are due to over-simplification of the assumed model, and that they may be corrected by (1) assuming a somewhat roughened ramp for the fault displacement time function to correspond to a stick-slip type of motion, and (2) assuming that the short period components of the fault displacement wave are coherent only over distances considerably smaller than the total fault length.

A matrix method for elastic wave problems

Abstract: 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.

Body force equivalents for seismic dislocations

Abstract: An explicit expression is derived for the body force to be applied in the absence of a dislocation, which produces radiation identical to that of the dislocation. This equivalent force depends only upon the source and the elastic properties of the medium in the immediate vicinity of the source and not upon the proximity of any reflecting surfaces. The theory is developed for dislocations in an anisotropic inhomogeneous medium; in the examples isotropy is assumed. For displacement dislocation faults, the double couple is an exact equivalent body force.

Surface waves in multilayered elastic media. I. Rayleigh and Love waves from buried sources in a multilayered elastic half-space

Abstract: A matrix formulation is used to derive integral expressions for the time transformed displacement fields produced by simple sources at any depth in a multilayered elastic isotropic solid half-space. The integrals are evaluated for their residue contribution to obtain surface wave displacements in the frequency domain. The solutions are then generalized to include the effect of a surface liquid layer. The theory includes the effect of layering and source depth for the following: (1) Rayleigh waves from an explosive source, (2) Rayleigh waves from a vertical point force, (3) Rayleigh and Love waves from a vertical strike slip fault model. The latter source also includes the effect of fault dimensions and rupture velocity. From these results we are able to show certain reciprocity relations for surface waves which had been previously proved for the total displacement field. The theory presented here lays the ground work for later papers in which theoretical seismograms are compared with observations in both the time and frequency domain.

Radiation pattern of surface waves from point sources in a multi-layered medium

Abstract: A previous treatment of the radiation pattern of Rayleigh waves from single and double couple point sources of arbitrary orientation in a semi-infinite homogeneous medium is generalized to the case of a medium composed of an arbitrary number of parallel homogeneous layers by the use of layer matrices. Source coefficients are given for a single force, a force dipole without torque, a single couple, and a double couple without torque, all for arbitrary orientation vectors, and also for a spherically symmetrical compressional source. For the simplest case permitting the existence of Love waves, a single finite layer over a semi-infinite medium, expressions for the Love wave radiation pattern are given explicitly in a form comparable to that previously given for Rayleigh waves in the homogeneous case.

Central U.S. crust—Upper mantle structure from Love and Rayleigh wave phase velocity inversion

Abstract: Fundamental mode Rayleigh and Love wave phase velocities in the period range 5 to 80 seconds have been measured in the central U. S. Using the concept of constant partial derivatives of phase velocity with respect to layer parameters in a least-squares inversion scheme a structural model for the area was determined. Several parameters were constrained on the basis of other available data. It was not possible to fit satisfactorily both Rayleigh and Love data with a simple isotropic model. Conclusions require either anisotropy in the upper mantle or systematic errors in the phase velocity measurements.

The seismicity of East Africa, the Gulf of Aden and the Arabian and Red Seas

Abstract: Maps of earthquake epicenters are presented for East Africa, the Gulf of Aden and the Arabian and Red Seas for the period January 1955 to March 1964. Many of these epicenters were located with an accuracy of about 10 km; errors of 100 km or more were common in previous studies. Several tectonic features can be resolved with the new epicenters. In the Gulf of Aden and in the Arabian Sea, the epicenters are confined to narrow linear segments. A large fracture zone that intersects the mid-oceanic ridge near 58°E is clearly delineated with the new epicenters. In the Gulf of Aden regions of high seismic activity are found where NNE-SSW trending faults intersect the median ridge. The seismic activity in the Red Sea is less than that in the Gulf of Aden and the Arabian Sea. A large number of the earthquakes in East Africa are associated with various branches of the rift system. However, many well-recorded earthquakes were not located along the rift valleys. The large areal extent of seismic activity in East Africa differs from the narrow linear pattern of activity that is associated with the mid-oceanic ridge.

Earth tides as a triggering mechanism for earthquakes

Abstract: Since earth tides represent the largest short-period oscillatory strains in the earth, a test has been made to see if any correlation exists between the times of occurrence of earthquakes in Southern California and the tidal potential. Two tests have been made, one of them a cross-correlation. On either basis, a statistically significant sample of earthquake events gives a correlation with the tidal potential that is of the same magnitude as a random sample.

Long period variations of the geomagnetic field

Abstract: The spectrum of fluctuations of the geomagnetic field extends from periods of < 10 years to periods of 106 years. Although cycles of constant period appear to be absent, definite segments of the spectrum may be associated with particular phenomena in the core. Secular variation with periods of 10 to 103 years is associated with changes in the nondipole field. Changes in the orientation and intensity of the main dipole contribute to the spectrum at about 104 years. At periods of 105 to 106 years, there may be a contribution from a random walk of the rotation axis. The more recent reversals of the polarity of the main dipole occur at characteristic intervals of about 106 years. The asymmetry of the nondipole component of the present geomagnetic field also appears in the results of new paleomagnetic studies of secular variation in the eastern Pacific basin. Two models for interpreting these results are presented, one based on lateral changes in the electrical conductivity of the lower mantle, and the other on the control of thermal convection patterns in the core by lateral temperature differences in the lower mantle.

The velocity of seismic waves near the earth's center

Abstract: A double velocity jump in the Earth's core entails a PKP travel-time curve with two lengthy branches extending back from 143°. The later branch is associated with the PKIKP phase. The earlier branch arises from waves, here designated PKHKP , which are refracted through the intermediate shell. Theoretical travel-time curves for PKP and SKS in possible Earth models with tripartite cores are presented. It is shown that the PKHKP branch provides an explanation for precursors to PKIKP observed at epicentral distances between 123° and 140°. Observations of waves predicted by the portion of this branch from 148° to 156° have been also reported. The SKS curve is examined in the light of some 550 SKS observations in the range 85° < Δ < 145°. The study provides evidence that there is in the core a discrete shell with thickness of order 420 kms and with a mean P velocity near 10.31 km/sec. This shell surrounds the inner core having mean radius 1220 kms and mean P velocity 11.22 km/sec, approximately. The material of the intermediate shell is not likely to have marked rigidity. The inner core is likely to be solid; published times for PKJKP waves may be, however, too small by several minutes.

Geographical interpretation of measurements of average phase velocities of surface waves over great circular and great semi-circular paths

Abstract: If the averages of the reciprocal phase velocity c −1 of a given Rayleigh or Love mode over various great circular or great semicircular paths are known, information can be extracted about how c −1 varies with geographical position. Assuming that geometrical optics is applicable, it is shown that if c −1 is isotropic its great circular averages determine only the sum of the values of c −1 at antipodal points and not their difference. The great semicircular averages determine the difference as well. If c −1 is anisotropic through any cause other than the earth9s rotation, even great semicircular averages do not determine c −1 completely. Rotation has negligible effect on Love waves, and if it is the only anisotropy present its effect on Rayleigh waves can be measured and removed by comparing the averages of c −1 for the two directions of travel around any great circle not intersecting the poles of rotation. Only great circular and great semicircular paths are considered because every earthquake produces two averages of c −1 over such paths for each seismic station. No other paths permit such rapid accumulation of data when the azimuthal variations of the earthquakes9 radiation patterns are unknown. Expansion of the data in generalized spherical harmonics circumvents the fact that the explicit formulas for c −1 in terms of its great circular or great semicircular integrals require differentiation of the data. Formulas are given for calculating the generalized spherical harmonics numerically.

Higher mode surface waves and their bearing on the structure of the earth's mantle

Abstract: A detailed numerical investigation of surface wave dispersion and particle motion associated with the higher Love and Rayleigh modes over realistic earth models has been carried out as a preliminary to the routine use of these waves in studies of the crust-mantle system. The suggestion that the so-called channel waves, such as the Lg, Li, and Sa phases, can be interpreted by higher mode group velocity dispersion curves is verified in detail. Furthermore, Sa should have a higher velocity across shield areas than across normal continental areas and a higher velocity across continents than across oceans. Higher mode Rayleigh wave data are presented for long oceanic paths to Pasadena. The observed data favor the CIT 11 model of Anderson and Toksoz (1963) over the 8099 model of Dorman et al. (1960) and indicate that under the Pacific Ocean the low-velocity zone extends to a depth perhaps as deep as 400 km followed by an abrupt increase in shear velocity.

Attenuation of shear waves in the upper and lower mantle

Abstract: The attenuation of seismic waves is a direct measure of the absorption due to nonelastic processes in the earth. The well known difficulties in obtaining body wave amplitude decrement data have been avoided by studying the spectral ratios of multiple ScS and sScS phases from two deep focus earthquakes recorded at near normal incidence. The average Q, for shear, in the mantle is about 600 for the frequency range 0.015 to 0.07 cps. Assuming that equal radiation occurs upwards and downwards from the source the average Q for the upper 600 km of the mantle is determined to be about 200 and about 2200 for the rest of the mantle. The value for Q at the base of the mantle is at least 5000 for shear waves.

The statistics of earthquakes in Southern California

Abstract: A statistically significant series of earthquakes in Southern California, 1934-1957, variously filtered to avoid aftershocks and events outside a localized area, has been tested for a Poisson distribution. It has been found that, for any of several methods of filtering, the data do not fit a Poisson distribution and hence a causal connection between weak earthquake events is inferred. The conclusion is in accord with that of Aki for large events.

The S wave project for focal mechanism studies earthquakes of 1962

Abstract: The Department of Geophysics of Saint Louis University has instituted a routine program for the determination of the focal mechanism of the larger earthquakes of each year using methods developed for the use of S waves in focal mechanism studies. Suites of records from selected stations are assembled from the WWSS microfilm file for each earthquake of interest. A combination of P-wave first motion and S-wave polarization data is then used to determine graphically the mechanism of the earthquakes. Thirty-six earthquakes of 1962 were selected for study. The focal mechanism solutions are presented for twenty-three of these shocks. There is evidence of patterns characteristic of the focal mechanism of earthquakes occurring in Kamchatka, the Aleutian Islands and South America. A complete presentation of all the data and of all the solutions is available in a more lengthy report.

Continuum distribution of dislocations on faults with finite friction

Abstract: An analysis is made of continuous distributions of infinitesimal dislocations on faults with finite friction. The analysis was undertaken in an attempt to explain the fact that dislocations produced by earthquakes commonly lie at depths that are shallower than the average depth of earthquake foci in continents. (The depths of dislocations are determined from displacements around faults.) It is found that this discrepancy can be explained if, at some depth, there exists a region where the frictional stress on faults is anomalously low.

The effect of boundaries on wave propagation in a liquid-filled porous solid: V. Transmission across a plane interface

Abstract: The passage of plane body waves across a plane interface from one to another, contiguous, porous aggregate is examined, with particular attention paid to motions involving wave lengths large in comparison with cross-sectional pore dimensions. The results are obtained for a rather general set of boundary conditions which take account of possible resistance to flow due to partial nonalignment of pores at the interface. It is found that when certain conditions of equality of material constants for the two media are met one or more of the reflected and transmitted waves are extinguished.

Observations of multiple seismic events

Abstract: 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.

Travel times, body waves, and normal modes of the earth

Abstract: This paper describes the physical relationship between the normal mode and ray representations of body waves, and develops equations and experimental procedures for analyzing body waves in terms of normal modes. For the simplest body waves considered, the horizontally-polarized shear waves S H and ( ScS ) H , the following equation relating the travel time curve to the phase velocity versus period curves corresponding to the various normal modes is derived: T = ( t − Δ d t / d Δ ) { I − 1 + [ ( T ) φ 2 − ( T ) φ 1 ] / 2 π } where T is the period corresponding to the phase velocity dΔ/ dt on the dispersion curve for the I th normal mode; t is the travel time corresponding to a distance Δ; and ϕ 1 ( T ) and ϕ 2 ( T ) are the spectral phases of a body wave at two successive points of reflection a distance Δ apart on a given ray path. The group velocity is given by: U = Δ t − { d [ ( T ) φ 2 − ( T ) φ 1 ] / d ω } I = const where ω is the angular frequency, 2π/ T . These equations are derived from a constructive phase interference requirement for successive reflections of body waves. They may also be applied to S v and PS waves over certain phase velocity ranges. T may be determined experimentally from the phase spectra of successive surface reflections of body waves of a given phase velocity (e.g., ScS at a distance Δ and ScSScS at a distance 2Δ). Analysis of S H and ( ScS ) H waves determines points on the torsional higher mode dispersion curves; and analysis of the S v or PS waves determines points on the spheroidal higher mode dispersion curves. Theoretical seismograms of multiply reflected ScS waves and actual seismograms of S , ScS and PS waves are analyzed to test and illustrate the method. Results are given for phase velocities ranging from 8 to 37 km/sec and for mode numbers ranging from 3 to more than 10. The limited results are in agreement with the shear velocity distribution of the Gutenberg model for a Caracas-Halifax path (Atlantic Ocean region) and with that of the Jeffreys model for a Bermuda-Alert path (partly Atlantic Ocean and partly Canadian shield). A method for rapid computation of approximate theoretical dispersion curves, based on the above equations, is presented.

The fine structure of the Earth's core

Abstract: Detailed study of arrivals from accurately fixed earthquakes has revealed additional complexity in the travel-time curve for PKP . A notation is introduced in which observations are denoted by P′ with a two-letter suffix indicating the branch to which they belong, namely P′ AB , P′ IJ , P′ GH and P′ DF . A new velocity solution for the Earth9s core has been derived from these observations. This velocity solution differs from those previously suggested in having three discontinuous increases in velocity between the outer and inner core, at levels corresponding to 0.570, 0.455 and 0.362 times the radius of the core. This implies two shells, each between 300 and 400 km thick, surrounding the inner core; in each shell there is a small negative velocity gradient. The outer discontinuity is sufficiently shallow to prevent rays in the outer core from forming a caustic.

Seismic waves with frequencies from 1 to 100 cycles per second recorded in a deep mine in Northern New Jersey

Abstract: A vertical component seismograph with a magnification of about 8 million at 20 cps was operated cps was operated 1780 feet below the surface at the Sterling mine of the New Jersey Zinc Company in Ogdensburg, New Jersey. Background noise, local earthquakes and teleseismic body phases were recorded. For frequencies greater than 2 cps, the continuous part of the background noise was fairly stationary during nighttime recording intervals and at a level lower than that at all but 9 of the 54 sites surveyed by Frantti (1963). During 730 hours of recording at quiet background noise levels, 8 local earthquakes were detected. Six local shocks were recorded during an additional 2160 hours by conventional short period instrumentation located at Ogdensburg and Sterling Forest, New York. These data, combined with historical data on the occurrence of 22 larger shocks local to the area, are consistent with a linear relation between the logarithm of earthquake number and magnitude. This relation is given by log N = 2.6−0.9M, where N is the annual number of earthquakes which occur within 300 km of Ogdensburg and which have magnitudes greater than or equal to M . On this basis the seismicity of the area, which lies within a region of Paleozoic and early Mesozoic diastrophism, is about 2 orders of magnitude less than that of Southern California, 3 orders of magnitude less than that of the Garm region in Central Asia, and 4 orders of magnitude less than that of the Kwanto region in Japan. Phases from 18 distant earthquakes were recorded by the high gain instrument. Only P and S phases from West Indies shocks, 20 degrees distant, exhibited motion with frequencies greater than 3 cps.

Travel times from central Pacific nuclear explosions and inferred mantle structure

Abstract: Travel time data, from widely recorded nuclear detonations in the Eniwetok and Bikini atolls of the central Pacific, have been compiled and are presented. Although a number of stations recorded ten or more events from each atoll, the resulting data may be considered as from a single point source, precisely known in time and place. Composite P-wave travel times are presented in a graphical form and, in the distance range from 3 to 102 degrees, are represented as eight near straight-line segments. P-wave speeds in the top of the mantle average about 8.2 km/sec to distances beyond 17 degrees, and a sharp discontinuity at 19.5 degrees is indicated. There is no evidence for or against a low-speed layer in the upper mantle nor for a regional shadow zone. A mantle model consisting of a number of discrete spherical shells has been constructed. A core depth of 2,870 km, 30 km short of the accepted value, is calculated from PcP arrival times at Matsushiro and College, which are 2.5 and 3.5 sec. earlier than are indicated in the Jeffreys-Bullen tables.

Universal dispersion tables 1. Love waves across oceans and continents on a spherical earth

Abstract: The variational approach to surface wave dispersion problems has been largely replaced by the powerful method of Haskell which is exact and particularly convenient for use on digital computers. This paper shows how the two methods may be combined to yield dispersion curves which can be used to interpret data from any layered structure. A set of graphs and tables is presented which can be used to calculate the dispersion of Love waves in the period range of 4 to 1000 seconds over any spherical earth model. In addition, it is possible to determine by inspection which portion of the earth is contributing to a set of observed dispersion data thereby facilitating the design of an appropriate earth model. These tables can be used to determine how much freedom can be taken with proposed models without violating dispersion data. Application to the inverse problem is immediate.

Determination of the natural periods of buildings

Abstract: Random wind excitation has been used to find the first three modes of vibration of a nineteenstory building about the two major translational axes and the torsional axis of the building. Vibration records were obtained with Willmore electromagnetic seismometers feeding into a multichannel magnetic tape recorder. A harmonic analysis was then performed with the aid of an analogue computer to determine the first few vibration modes. Concurrently with the experimental program the building modes were computed two ways using different simplifying assumptions regarding the lateral stiffness of the structure.

On the travel times of P as determined from nuclear explosions

Abstract: A study has been made of the travel times of P waves as recorded from 14 shots on the Nevada Test Site (NTS) and from the Gnome (GN) shot. The travel times on different lines extending from the test sites were considered separately. Of special interest were the travel times obtained on the line joining the NTS and GN site. The two sets of travel times were in good agreement, but from epicentral distance 700 km onwards they were delayed and scattered. This was an indication of the presence of a low velocity layer at small depth on the line. Travel times on lines extending in other directions from the NTS differed somewhat but not very greatly from those on the NTS-GN line. On the line from the Gnome site to the northeast travel times are considerably smaller owing to a strong increase of velocity with depth close to the Gnome site. Travel times in agreement with those observed were calculated. Some California stations at distances around 1700 km observed P of the Gnome shot with an exceptional delay of 4 1 2 seconds. The new Madrid earthquake of February 2, 1962, had its epicenter on the Gnome northeast line, but the surface velocity of Pn of the earthquake is smaller on the line than that of the Gnome shot. Pn of the earthquake was well recorded in all directions from the epicenter; there is no low velocity layer at small depth in these regions. In an earlier study of some earthquakes in northeastern America Pn and also Sn were found to be well recorded phases. It may seem as if shallow, low-velocity layers are confined to the western mountain regions.

The tectonic stress and tectonic motion direction in Europe and Western Asia as calculated from earthquake fault plane solutions

Abstract: The statistics of fault plane solutions of earthquakes is further analyzed and it is shown that, to find a best axis or best plane to a set of axes , the eigenvectors of a certain matrix must be calculated. The justification for this procedure follows from the same argument as that of Fisher who showed that the best of a series of directions is obtained by forming the vector sum. The eigenvector technique is then applied to the pertinent axes of fault plane solutions of earthquakes that occurred in Europe and Western Asia. It is shown that, in this region, the focal mechanisms of the earthquakes tend to orient themselves in such a fashion that the P axes coincide with the principal horizontal stress directions, the latter being normal to the geographically prominent features. The null axes tend to lie in a plane normal to the best fitting P axes. The chief random element enters into the orientation of the T axes. All this is in conformity with the predictions of theory.

Arrival times of P waves and upper mantle structure

Abstract: Residuals of arrival times of P waves, as given in the International Seismological Summary for Kiruna, Sweden, Reykjavik, Iceland, and Scoresbysund, Greenland, were studied in order to detect upper mantle anomalies. The Kiruna arrivals were systematically too early, with a mean residual of −1.4 seconds, while the mean Reykjavik residual was +1.3 seconds. The difference in mean residual was 2.7 seconds with a standard error of about 0.5 second. The mean residual at Scoresbysund was −0.4 second. It is assumed that there is a depth D below which the mantle is homogeneous. The difference in mean residuals at the stations is assumed to be caused by different wave velocities at depths less than D in the vicinities of the stations. If it is assumed that the P -wave velocity in the upper mantle is constant down to a depth D below each station, this depth can be computed. This velocity is known from other data to be 8.36 km/sec below Kiruna and 7.4 km./sec. below Reykjavik. If only earthquakes at distances from 20° to 39° were used, D is determined to be 246 ± 36 km. (standard error). Earthquakes at distances 40° to 59° give D = 177 ± 25 km., at distances 60° to 79° give D = 234 ± 14 km., and at distances 80° to 99° give D = 281 ± 20 km. The most probable value of D is thus about 240 km. below the earth9s surface, with a standard error of about 40 km. In the vicinity of Scoresbysund the upper mantle velocity is found to be about 8.0 km./sec., using the same assumption.

The effect of boundaries on wave propagation in a liquid-filled porous solid: VI. Love waves in a double surface layer

Abstract: The classical solution of Stoneley and Tillotson is generalized by considering the outer one of the pair of layers to be porous. Although the dispersion relation turns out, for practical purposes, to be identical with the one governing the classical case, the motion in the present instance is shown to be dissipative and the expression is exhibited for the specific energy loss.

On a method to obtain a Green's function for a multi-layered half space

Abstract: In this paper the surface wave terms of the Green9s function for a two-dimensional multilayered half space are obtained. The method used is new and remarkable by its simplicity. It is based on the integral representation theorems for elastodynamics. The orthogonality properties of surface waves are generalized to include not only Love waves but Rayleigh waves as well.