Publisher Summary This chapter focuses on elastic wave propagation in stratified media. The development of the theory of elastic wave propagation in stratified media has been strongly influenced by the problems of seismic wave propagation and the nature of the seismograms recorded from earthquakes. For purely analytic developments of elastic wave propagation, the level of manageable algebraic complexity is reached in a model with one or two uniform layers overlying a uniform half space. This chapter shows how the excitation of elastic waves, within a horizontally stratified structure, can be conveniently developed in terms of reflection and transmission matrices. This procedure has allowed the construction of the full response of the medium or approximations with desired properties so that theoretical seismograms may be calculated for realistic distributions of elastic parameters. Although this development has been for isotropic media, nearly all the results apply directly to the case of full anisotropy if 3 × 3 reflection and transmission matrices allowing coupling between all wave types are employed. This development of the wavefield for both source and receiver within the stratification may be used for other classes of wave propagation.

Seismic Wave Propagation in Stratified Media

Summary. The response of a stratified elastic half space to a general source may be represented in terms of the reflection and transmission properties of the regions above and below the source. For P-SV and SH waves and both buried sources and receivers, convenient forms of the response may be found in which no loss of precision problems arise from growing exponential terms in the evanescent regime. These expressions have a ready physical interpretation and enable useful approximations to the response to be developed. The reflection representation leads to efficient computational procedures for models composed of uniform layers, which may be extended in an asymptotic development to piecewise smooth models.

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Seismic waves in a stratified half space.

The connection is established between conventional matrix methods for layered media and the reflection and transmission properties of a single layer. This interrelation is then used to set up an iterative approach to the calculation of reflection and transmission coefficients in multilayered media. This approach lends itself to a ray interpretation and allows estimates of errors involved in taking truncated partial ray expansions to be made. The special effects due to a free surface are also considered.

Reflections, rays, and reverberations

In a porous material saturated by a fluid electrolyte, mechanical and electromagnetic (EM) disturbances are coupled. The coupling is electrokinetic in nature. The seismic waves generate relative fluid-solid motion that induces an electrical streaming current. When a seismic pulse traverses contrasts in elastic and/or fluid-chemistry properties, the streaming-current imbalance creates dipolar and multipolar charge separations across the interface that, in turn, produce EM disturbances that are measurable at the earth's surface. This paper numerically determines a full-waveform electroseismic point-source response in a stratified porous medium. It is shown that the macroscopic governing equations controlling the coupled electromagnetics and acoustics of porous media decouple into two sets corresponding to vertical or horizontal polarization of the transverse wave fields. The frequency content of the converted EM field has the same frequency content (at the generating interface) as the incident seismic pulse. Snapshots in time and converted EM amplitudes versus source to antenna offset are calculated for contrasts in mechanical and/or electrical medium properties. The converted EM radiation pattern away from the interface is similar to having an effective vertical-electric dipole centered right beneath the source on the contrast. The transverse magnetic mode amplitudes fall off rapidly with distance, from the generating interface thus suggesting the importance of a vertical electroseismic profiling geometry to record the converted EM signal at antennas close to an interface of interest.

Electroseismic waves from point sources in layered media

Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmetry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmetry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.

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Seismic waves in stratified anisotropic media

Summary

The aim of the series of papers to be published under the above title is to follow up and improve Carpenter's (1966) attempt to construct realistic pulse shapes for teleseismic body waves from underground explosions. The present work deals with both explosions and shallow earthquakes and with the radiation of surface waves as well as body waves.



The extension of Carpenter's theory to take crustal layering more accurately into account involves the matrix theory first introduced by Thomson (1950) and developed by Haskell. We shall use the notation of Haskell's (1964) paper in the following analysis and the initial theoretical work is concerned with some of the groundwork and one or two results which have not been covered by Haskell in his series of papers (1953, 1962, 1964).



A point source can be represented either as a system of forces or as a discontinuity in the displacement or stress or their derivatives across an element of surface. We show here that a general source of either type is equivalent in the generation of elastic radiation to a discontinuity across a horizontal plane in the displacement and the stress acting on the plane. This means that any point source can be put into a form suitable for computations based on the Thomson-Haskell theory.



The theory is applied to the construction of theoretical models of earthquake and explosive sources. Some of the more realistic models so far proposed are given in the later section of the paper.

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J. A. Hudson

Papers

A Quantitative Evaluation of Seismic Signals at Teleseismic Distances—I Radiation from Point Sources