Mode-ray duality

Earlier results in the theory of terrestrial radio waves are applied to seismology. A partial field of the complete eigen-value solution for a sphere can be interpreted as real rays. Watson's transformation and the WKBJ approximation are employed to establish links between the index trio (l, m, n) of a mode and the corresponding parameters of the ray trajectory associated with this mode. It is shown that Snell's law for rays and Jean's formula are complementary. The condition of constructive interference is expressed as an integral equation for the eigen-frequencies _nω_l

Spectral response of an elastic sphere to dipolar point-sources

A stratified elastic sphere is excited by an harmonic dipolar source of arbitrary orientation and depth. The total field is expanded in series of vector spherical harmonics and then condensed into a convenient form of a displacement dyadic. The Haskell-Gilbert matrix method is employed to obtain the radial factor of the displacements for a multilayered sphere. The dependence of the field on the azimuth angle and the fault elements is obtained for the case of a double-couple at depth. Expressions are also developed for the radiation pattern of surface waves over a spherical stratified earth

The Equivalent Force System Of A Monopole Source In A Fluid-Filled Open Borehole

The elastodynamic body-wave field outside a fluid-filled open borehole due to a monopole source in the fluid, is reduced to the radiation-field due to a suitable equivalent force system (EFS) in the absence of the borehole, consisting of a monopole plus a vertical dipole. Theoretical seismograms of the EFS displacements in the solid are shown to be in excellent agreement with those obtained from the exact solution to the fluid-filled open borehole problem.United States. Defense Advanced Research Projects Agency (Contract F1962889-K-0020

Source-mechanism from spectra of long-period seismic surface waves. 3. The Alaska earthquake of July 10, 1958

Source-mechanism is derived from amplitude and phase spectra of mantle Love and Rayleigh waves of the Alaska earthquake of July 10, 1958. The signals R_2, R_3, G_2, G_4, G_5 recorded on the Gilman 80–90 and the Press-Ewing 30–90 seismograph systems at Pasadena, California, are separated, digitized, filtered and Fourier-analyzed. An agreement between theory and observations is obtained for a unilateral fault of 300–350 km, which ruptured with a speed of 3-3.5 km/sec in the direction N40°W. Fault length is in good agreement with the extent of aftershock distribution in the month of July, 1958, and the time of rupture checks with the duration of an impressive T-phase recorded at Hawaii. The phases of the signals are corrected for propagation, instrumental shift and the source finiteness. Initial phases thus obtained agree on a mechanism of a right double-couple with a unit step-function in time

Velocities of mantle Love and Rayleigh waves over multiple paths

Phase velocities of Love waves from five major earthquakes are measured over six great circle paths in the period range of 50 to 400 seconds. For two of the great circle paths the phase velocities of Rayleigh waves are also obtained. The digitized seismograph traces are Fourier analyzed, and the phase spectra are used in determining the phase velocities. Where the great circle paths are close, the phase velocities over these paths are found to be in very good agreement with each other indicating that the measured velocities are accurate and reliable. Phase velocities of Love waves over paths that lie far from each other are different, and this difference is consistent and much greater than the experimental error. From this it is concluded that there are lateral variations in the structure of the earth's mantle. One interpretation of this variation is that the mantle under the continents is different from that under the oceans, since the path with the highest phase velocities is almost completely oceanic. This interpretation, however, is not unique and variations under the oceans and continents are also possible. Group velocities are computed from the phase velocities and are also directly measured from the seismograms. The group-velocity curve of Love waves has a plateau between periods of 100 and 300 seconds with a shallow minimum at about 290 seconds. The sources of error in both Fourier analysis and direct time domain methods of phase velocity measurement are discussed

Radiation patterns of seismic surface waves from buried dipolar point sources in a flat stratified Earth

Explicit compact expressions were obtained for the far displacement field of Rayleigh and Love waves generated by force configurations which served to simulate shear-type faults with arbitrary dip and slip. The medium transfer functions for dipolar sources were computed for a Gutenberg flat continental earth model with 23 layers. These were then used to obtain universal radiation pattern charts for couple- and double-couple-type sources at various depths over the period range 50 to 350 sec. It was demonstrated by means of few typical examples that the radiation patterns of Rayleigh waves may depend strongly on the depth of the source, and unlike the fundamental Love mode may be rather sensitive to small variations in frequency. For a given source and frequency the radiation pattern may differ considerably from one mode to another

Source mechanism from spectrums of long-period surface waves: 2. The Kamchatka earthquake of November 4, 1952

Fourier analysis of mantle Love and Rayleigh waves from the source of the Kamchatka earthquake of November 4, 1952, recorded on the Benioff linear strain seismograph at Pasadena, furnished further evidence in support of the moving-source theory. Amplitude and phase spectrums of G_1, G_2, G_3, G_4, R_2, and R_3 were processed to obtain information on the mechanism at the source. Both the directivity and the differential phase agree on a unilateral fault of 700 km which ruptured with a speed of 3 km/sec in the direction N 146° W. The fault length is in good agreement with the extent of aftershock distribution in the month of November 1952. The initial phases of Love and Rayleigh waves agree on a mechanism of a right orthogonal double couple with a time dependence which is close to the Heaviside step function

Fast evaluation of source parameters from isolated surface-wave signals. Part I. Universal tables

Tables for spectral displacements of seismic surface waves from shear dislocations in flat multilayered earth models were prepared. Earth response functions for seven modes (R_(11), R_(21), R_(12), L_0, L_1, L_2, L_3) at six periods (300 sec, 250 sec, 200 sec, 150 sec, 100 sec, 50 sec) and three paths (continental, oceanic, shield) were calculated for the source-depth range of 10 to 600 km at intervals of 5 km until 200 km, and thereafter at intervals of 10 km. Ground motion is given in micron-seconds for the three fundamental shear dislocations, each of strength U_0dS = 10^3 (m × km^2) and a delta-function time-dependence. The tables provide the means for rapid evaluation of source parameters from spectral radiation patterns of amplitudes and initial phases

Determination of source parameters by amplitude equalization of seismic surface waves: 2. Release of tectonic strain by underground nuclear explosions and mechanisms of earthquakes

The radiation patterns of Love and Rayleigh waves from three nuclear explosions (Hardhat, Haymaker, and Shoal) are studied to determine the nature of the asymmetry of radiation and the mechanism of Love wave generation. From a comparative study of different explosions it is reasoned that the Love waves are generated at the source of the explosion. The source function, represented as the superimposition of an isotropic dilatational component due to the explosion and a multipolar component due to the release of tectonic strain energy, is consistent with the observed radiation patterns and the amplitude spectrums. The amount of seismic energy due to the strain release is computed. In some cases (Haymaker and Shoal) it is found that this energy may be due to the relaxation of the pre-stressed medium by the explosion-formed cavity. In the case of Hardhat it is concluded that the explosion must have triggered some other strain release mechanism, such as an earthquake. The amplitude equalization method is applied to surface waves from an earthquake to determine the source parameters. From only the amplitude spectrums and radiation patterns of Love and Rayleigh waves, the source functions, source depth, strike and dip of the fault plane, and the rake of displacement are determined for the July 20, 1964, Fallon earthquake