Reply [to “Comments on paper by David G. Harkrider, ‘Theoretical and observed acoustic-gravity waves from explosive sources in the atmosphere’”]

I acknowledge and agree with the comments made on Harkrider [1964] by A.D. Pierce

Seismic representation theorem coupling: synthetic SH mode sum seismograms for non-homogeneous paths

In this paper the methods for representation theorem coupling of finite-element or finite difference calculations and propagator matrix method calculations (Harkrider) are developed.The validity and accuracy of the resulting hybrid method are demonstrated.The resulting hybrid technique can be used to study the propagation of any phase that can be represented in terms of an SH mode sum seismogram, across regional transition zones or other heterogeneities. These heterogeneities may exist in regions which form subsegments of a longer, mostly plane-layered, path. Examples of structures of interest through which such waves can be propagated using these techniques include, regions of crustal thickening or thinning such as continent-ocean transitions or basins, anomalous bodies of any shape located in the path, and sudden transitions from one layered structure to another. Examples of the types of phases that may be propagated through these structures include Love waves, L_g, S_n, and S_a

Seismic Source Descriptions of Underground Explosions and a Depth Discriminate

Synthetic seismograms of both body waves and Rayleigh waves are used to determine the radiation field of a few large contained underground explosions. A number of possible source descriptions are investigated. A reduced displacement potential of the form, ø(t) = ø_0t^ξ exp(-ηt), fits the long- and short-period data. The source parameters appropriate for the Boxcar event are ξ = 0·5 and η = 0·15. Synthetic PL and Rayleigh waves are compared with observations from a number of different size events to determine the dependence of η on yield. The amplitude of the long period synthetic body wave responses at ranges greater than about 12° increases rapidly as the source depth is increased. Thus the difference in spectral properties of explosions and earthquakes can be largely explained by the depth effect. The theoretical ratio SP/LP, that is the short period divided by the long-period amplitude, is computed from 12 to 25° for the Johnson upper mantle model and the Boxcar source. A study of an earthquake which cannot be distinguished from an explosion using the m_b vs. M_s criterion is investigated by the SP/LP discriminate

Numerical modelling of SH L_g waves in and near continental margins

The effect of transition regions between continental and oceanic structures on the propagation of L_g waves from continental sources is examined. In particular, the attenuation due to variations in layer thickness in such transition regions is calculated and explained for a suite of simple models. The measured attenuation, due to the geometry of the transition regions between the oceanic and continental structures within a partially oceanic path with source and receiver in a continental structure, is at most a factor of four for frequencies from 0.01 to 1 Hz. This is inadequate to explain the observed extinction of L_g along such paths. This extinction has previously been attributed to the effects of the transition region geometry. The method used to calculate the results presented in this study is developed and its validity and accuracy are demonstrated. Propagator matrix seismograms are coupled into a Finite Element calculation to produce hybrid teleseismic SH mode sum seismograms. These hybrid synthetics can be determined for paths including any regional transition zone or other heterogeneity that exists as part of a longer, mostly plane-layered, path. Numerical results presented for a suite of transition models show distinct trends in each of the regions through which the wavefield passes. The wavefield passes through a continent-ocean transition region, then a region of oceanic structure, and finally through an ocean-continent transition region. When an L_g wavefront passes through a continent-ocean transition, the amplitude and coda duration of the L_g wave at the surface both increase. At the same time, much of the modal L_g energy previously trapped in the continental crust is able to escape from the lower crust into the subcrustal layers as body waves. The magnitude of both these effects increases as the length of the transition region increases. When the wavefront passes through the region of oceanic structure further energy escapes from the crustal layer, and produces a decrease in L_g amplitude at the surface. The rate of amplitude decrease is maximum near the transition region and decreases with distance from it. When the wavefield passes through the ocean-continent transition region a rapid decrease in the L_g amplitude at the surface of the crust results. The energy previously trapped in the oceanic crustal layer spreads throughout the thickening crustal layer. Some of the body wave phases produced when the wavefield passes through the continent-ocean transition region are incident on the continental crust in the ocean-continent transition region. These waves are predominantly transmitted back into the crust. The other body wave phases reach depths below the depth of the base of the continental crust before reaching the ocean-continent transition and, thus, escape from the system

A note on the existence of relative maxima and minima on phase velocity curves

Phase and group velocity dispersion curves for fundamental Rayleigh waves have been computed with more precision than previously attempted. The new curves show a relative minimum in phase velocity at periods near 50 sec for four perturbed Gutenberg continental models

A generalized reflection-transmission coefficient matrix and discrete wavenumber method for synthetic seismograms

Expressions for displacements on the surface of a layered half-space due to point force are given in terms of generalized reflection and transmission coefficient matrices (Kennett, 1980) and the discrete wavenumber summation method (Bouchon, 1981). The Bouchon method with complex frequencies yields accurate near-field dynamic and static solutions. The algorithm is extended to include simultaneous evaluation of multiple sources at different depths. This feature is the same as in Olson's finite element discrete Fourier Bessel code (DWFE) (Olson, 1982). As numerical examples, we calculate some layered half-space problems. The results agree with synthetics generated with the Cagniard-de Hoop technique, P-SV modes, and DWFE codes. For a 10-layered crust upper mantle model with a bandwidth of 0 to 10 Hz, this technique requires one-tenth the time of the DWFE calculation. In the presence of velocity gradients, where finer layering is required, the DWFE code is more efficient

Precision of the determination of focal depth from the spectral ratio of Love/Rayleigh surface waves

The precision with which the focal depth may be determined using Love/Rayleigh-wave spectral ratios depends on the accuracy of the models for Earth structure and for source mechanism used in the focal depth calculations. Estimates of the precision of the focal depth determination are obtained using the partial derivatives of Love/Rayleigh spectral ratios with respect to the parameters: focal depth, shear velocity, dip angle, and slip angle. We find that errors caused by imprecise knowledge of any of these parameters can be important in practice

Theoretical Effect of Yield and Burst Height of Atmospheric Explosions on Rayleigh Wave Amplitudes

Theoretical seismograms for fundamental mode Rayleigh waves were calculated for atmospheric point sources over oceanic and over continental Earth models, as recorded at an epicentral distance of 10000 km. Yields were uniformly distributed over the range 1 kT-10 MT, for source altitudes in the range 0.3-92.0 km. The Earth structures used were those of Gutenberg and of Anderson and Toksöz. The source models were point mass-injection and energy-injection sources at altitude, as well as a distributed pressure pulse at the surface of the Earth. It was found that: (1) as far as Rayleigh wave excitation is concerned, the mass-injection and energy-injection sources are equivalent; (2) for low altitudes the Rayleigh wave excitation is independent of source type, but at intermediate altitudes the surface overpressure source predicts greater amplitudes than the other two source models; (3) for most altitudes, the energy coupling from the atmosphere into Rayleigh waves is more efficient for the continental Earth structure than for the oceanic structure; (4) Rayleigh wave amplitude is more sensitive to yield than to burst height (5) dependence of Rayleigh wave amplitude is less than the cube root relation for low-yield explosions at intermediate altitudes but greater for high-yield explosions at near-surface altitudes; (6) spectral splitting ratios do not show a systematic variation with yield and burst height

Determination of source parameters of explosions and earthquakes by amplitude equalization of seismic surface waves: 1. Underground nuclear explosions

A method of determining the source parameters of explosions and earthquakes from the amplitude spectrums of seismic surface waves is described. The method, called amplitude equalization, involves the correction of the ground displacement spectrum for the propagation effect. This is accomplished by multiplying it numerically with the inverse of the frequency response of the layered medium. The result is the amplitude spectrum of the source function, which may be interpreted by itself or jointly with the initial phase spectrum to determine the source-time variation. The spectrums of the Rayleigh waves from underground nuclear explosions are compared and the source-time function is interpreted using the amplitude equalization method. The time variation of the pressure pulse at the boundary of the elastic zone is found to be of the form p(t) = P_0te^(−ηt), where η is a parameter which depends on the yield of the explosion and on the medium. For the events studied, the breadth of the pulse increased (η decreased) with the yield of the explosion