Seismic body waves in anisotropic media: reflection and refraction at a plane interface

Summary. A formulation is derived for calculating the energy division among waves generated by plane waves incident on a boundary between generally anisotropic media. A comprehensive account is presented for P, SV and SH waves incident from an isotropic half-space on an orthorhombic olivine half-space, where the interface is parallel to a plane of elastic symmetry. For comparison, a less anisotropic medium having transverse isotropy with a horizontal axis of symmetry is also considered. The particle motion polarizations of waves in anisotropic medium differ greatly from the polarizations in isotropic media, and are an important diagnostic of the presence of anisotropy. Incident P and SV waves generate quasi- SH waves, and incident SH waves generate quasi- P and quasi- SV waves, often of considerable relative magnitude. The direction of energy transport diverges from the propagation direction.     

Receiver functions at the stations of the German Regional Seismic Network (GRSN)

Several years of broad-band teleseismic data from the GRSN stations have been analysed for crustal structure using P -to- S converted waves at the crustal discontinuities. An inversion technique was developed which applies the Thomson-Haskell formalism for plane waves without slowness integration. The main phases observed are Moho conversions, their multiples in the crust, and conversions at the base of the sediments. The crustal thickness derived from these data is in good agreement with results from other studies. For the Gräfenberg stations, we have made a more detailed comparison of our model with a previously published model obtained from refraction seismic experiments. The refraction seismic model contains boundaries with strong velocity contrasts and a significant low-velocity zone, resulting in teleseismic waveforms that are too complicated as compared to the observed simple waveforms. The comparison suggests that a significant low-velocity zone is not required and that internal crustal boundaries are rather smooth.     

Polarization and amplitude attributes of reflected plane and spherical waves

The characteristics of a reflected spherical wave at a free surface are investigated by numerical methods; in particular, the polarization angles and amplitude coefficients of a reflected spherical wave are studied. The classical case of the reflection of a plane P wave from a free surface is revisited in order to establish our terminology, and the classical results are recast in a way which is more suited for the study undertaken. The polarization angle of a plane P wave, for a given angle of incidence, is shown to be 90° minus twice the angle of reflection of the reflected S wave. For a Poisson's ratio less than 1/3, there is a non-normal incident angle for which both amplification coefficients are 2 precisely; for this incident angle the direction of the particle motion at the free surface is also the direction of the incident wave. For a wave emanating from a spherical source, the polarization angle, for all angles of incidence, is always less than, or equal to, the polarization angle of a plane P wave. The vector amplification coefficient of a spherical wave, for all angles of incidence, is always greater than the vector amplification coefficient of a plane P wave. As expected, the results for a spherical wave approach the results for a plane P wave in the far field. Furthermore, there was a good agreement between the theoretical modelling and the numerical modelling using the dynamic finite element method (DFEM).     

Shear-wave polarizations on a curved wavefront at an isotropic free surface

Summary. We present polarization diagrams of the particle motions at the free surface of an isotropic half-space generated by incident shear waves from a local buried point source. The reflectivity technique is used to calculate synthetic seismograms from which the particle motions are plotted. The particle motions are examined over a range of epicentral distances in a uniform isotropic half-space for different source frequencies and polarization angles, and for different Poisson's ratios. The particle motions due to a curved wavefront possess different characteristics from those generated by plane wavefronts at corresponding incidence angles. A curved wavefront generates a local SP -phase: a P -headwave which propagates along the free surface, and arrives shortly before the direct S -wave. These two arrivals give rise to cruciform particle motions in the sagittal and horizontal planes, which could be misinterpreted as anisotropy-induced shear-wave splitting. An examination of the particle motion in the transverse plane, mutually orthogonal to the sagittal and horizontal planes, can be used to discriminate between isotropic and anisotropic interpretations. The amplitude of the SP -phase is enhanced when it propagates in a low-velocity surface layer overlying the source layer, and may then become the dominant phase on radial-component seismograms. The presence of even a single surface layer may introduce considerable complexity into the seismogram, and we examine the effects of layer thickness, velocity contrast, and source depth on the corresponding polarization diagrams. Reliable information on the source and propagation path characteristics of shear waves from a buried local point source can only be obtained from free-surface records if they are recorded within a very limited epicentral distance range.     

Diffraction of P, S and Rayleigh waves by three-dimensional topographies

The diffraction of P, S and Rayleigh waves by 3-D topographies in an elastic half-space is studied using a simplified indirect boundary element method (IBEM). This technique is based on the integral representation of the diffracted elastic fields in terms of single-layer boundary sources. It can be seen as a numerical realization of Huygens principle because diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. A Fredholm integral equation of the second kind for such sources is obtained from the stress-free boundary conditions. A simplified discretization scheme for the numerical and analytical integration of the exact Green's functions, which employs circles of various sizes to cover most of the boundary surface, is used.
The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves.     

Scattering of waves in elastic media containing passive and active cracks

Scattering of wavefields in a 3-D medium that includes passive and/or active structures, is numerically solved by using the boundary integral equation method (BIEM). The passive structures are velocity anomalies that generate scattered waves upon incidence, and the active structures contain endogenous fracture sources, which are dynamically triggered by the dynamic load due to the incident waves. Simple models are adopted to represent these structures: passive cracks act as scatterers and active cracks as fracture sources. We form cracks using circular boundaries, which consist of many boundary elements. Scattering of elastic waves by the boundaries of passive cracks is treated as an exterior problem in BIEM. In the case of active cracks, both the exterior and interior problems need to be solved, because elastic waves are generated by fracturing with stress drop, and the growing crack boundaries scatter the incident waves from the outside of the cracks. The passive cracks and/or active cracks are randomly distributed in an infinite homogeneous elastic medium. Calculations of the complete waveform considering a single scatter show that the active crack has weak influence on the attenuation of first arrivals but strong influence on the amplitudes of coda waves, as compared with those due to the passive crack. In the active structures, multiple scattering between cracks and the waves triggered by fracturing strongly affect the amplitudes of first arrivals and coda waves. Compared to the case of the passive structures, the attenuation of initial phase is weak and the coda amplitudes decrease slowly.     

Scattering of surface waves modelled by the integral equation method

The integral equation method is used to model the propagation of surface waves in 3-D structures. The wavefield is represented by the Fredholm integral equation, and the scattered surface waves are calculated by solving the integral equation numerically. The integration of the Green's function elements is given analytically by treating the singularity of the Hankel function at   R = 0  , based on the proper expression of the Green's function and the addition theorem of the Hankel function. No far-field and Born approximation is made. We investigate the scattering of surface waves propagating in layered reference models imbedding a heterogeneity with different density, as well as Lamé constant contrasts, both in frequency and time domains, for incident plane waves and point sources.     

Seismic vertical array analysis for phase decomposition

We propose a vertical array analysis method that decomposes complex seismograms into body and surface wave time histories by using a velocity structure at the vertical array site. We assume that the vertical array records are the sum of vertically incident plane P and S waves, and laterally incident Love and Rayleigh waves. Each phase at the surface is related to that at a certain depth by the transfer function in the frequency domain; the transfer function is obtained by Haskell's matrix method, assuming a 1-D velocity structure. Decomposed P , S and surface waves at the surface are estimated from the vertical array records and the transfer functions by using a least-squares method in the frequency domain; their time histories are obtained by the inverse Fourier transform. We carried out numerical tests of this method based on synthetic vertical array records consisting of vertically incident plane P and S waves and laterally incident plane Love and Rayleigh waves. Perfect results of the decomposed P , S , Love and Rayleigh waves were obtained for synthetic records without noise. A test of the synthetic records in which a small amount of white noise was added yielded a reasonable result for the decomposed P , S and surface waves. We applied this method to real vertical array records from the Ashigara valley, a moderate-sized sedimentary valley. The array records from two earthquakes occurring at depths of 123 and 148 km near the array (epicentral distance of about 31 km) exhibited long-duration later phases. The analysis showed that duration of the decomposed S waves was a few seconds and that the decomposed surface waves appeared a few seconds after the direct S -wave arrival and had very long duration. This result indicated that the long-duration later phases were generated not by multireflected S waves, but by basin-induced surface waves.     

Guided waves in three-dimensional structures

A new formulation for the propagation of surface waves in three-dimensionally varying media is developed in terms of modal interactions. A variety of assumptions can be made about the nature of the modal field: a single set of reference modes, a set of local modes for the structure beneath a point, or a set of local modes for a laterally varying reference structure. Each modal contribution is represented locally as a spectrum of plane waves propagating in different directions in the horizontal plane. The influence of 3-D structure is included by allowing coupling between different modal branches and propagation directions. For anisotropic models, with allowance for attenuation, the treatment leads to a set of coupled 2-D partial differential equations for the weight functions for different modal orders.
The representation of the guided wavefield requires the inclusion of a full set of modes, so that, even for isotropic models, both Love and Rayleigh modes appear as different polarization states of the modal spectrum. The coupling equations describe the interaction between the different polarizations induced by the presence of the 3-D structure.
The level of lateral variation within the 3-D model is not required to be small. Horizontal refraction or reflection of the surface wavefield can be included by allowing for transfer between modes travelling in different directions. Approximate forms of the coupled equation system can be employed when the level of heterogeneity is small, for example the coupling between the fundamental mode and higher modes can often be neglected, or forward propagation can be emphasized by restricting the interaction to a limited band of plane waves covering the expected direction of propagation.     

Propagation of harmonic plane waves in a general anisotropic porous solid

Wave propagation is studied in a general anisotropic poroelastic solid. The presence of dissipation due to fluid-viscosity as well as hydraulic anisotropy of pore permeability are also considered. Biot's theory is used to derive a system of modified Christoffel equations for the propagation of plane harmonic waves in porous media. A non-trivial solution of this system is ensured by a determinantal equation. This equation is separated into two different polynomial equations. One is the quartic equation whose roots represent the complex velocities of four attenuating waves in the medium. The other is a eighth-degree polynomial whose roots represent the vertical slowness values for the four waves propagating upward and downward in a finite porous medium. Procedure is explained to associate the numerically obtained roots with the waves propagating in the medium. The slowness surfaces of waves reflected at the boundary of the medium are computed for a realistic numerical model. The behaviours of phase velocity surfaces are analysed with the help of numerical examples.     

Wave propagation characteristics in frozen saturated soil

Ultrasonic detection technology is of great significance in the detection and evaluation of physical and mechanical properties of frozen soil, but wave propagation characteristics in frozen soil are unclear. Based on the three-phase composition of frozen saturated soil and the mixture theory, considering Bishop's effective stress formula, the wave propagation equations are establish for frozen saturated soil. In wave propagation, an entropy inequality was introduced to describe the coupling of different phases. The analytic expressions of propagation velocity and attenuation law of waves in frozen soil are obtained, and wave propagation characteristics in frozen saturated soil are discussed. Results show that four types of waves(i.e., P1, P2, P3 and S) are found in frozen saturated soil and all four wave types are dissipative waves, in which the attenuation of P3 is the maximum. The velocity of four waves increases sharply at the excitation frequency range of 10~3–10~9 Hz,but the wave velocity at high-frequency and low-frequency is almost constant. When volume ice content increases, the wave propagation velocity of P1 and S decreases dramatically, and the velocity of P2 increases gradually, but P3 velocity increases first and then decreases to zero with increasing saturation. The attenuation coefficients of P1 and S waves begins to increase gradually when the volume ice content is about 0.4, P2 increases first and then decreases with an increase of volume ice content and P3 increases with the volume ice content and decreases rapidly from extreme to zero.     

Interaction of a compressional impulse with a slot normal to the surface of an elastic half space

Summary. An improved finite difference scheme is applied to simulate wave propagation in the vicinity of a slot normal to the surface of an elastic half space. It provides visualization of the scattered wave pattern at a sequence of time steps, and also the components of displacement as functions of time at a series of observation points.
After being hit by a normally incident plane P pulse, the slot oscillates with two main cycles and two shear-compressional pairs of diffracted waves, and also Rayleigh pulses, are scattered from it. The resulting wavefronts are parallel to the vertical surfaces of the slot and curve in semicircular arcs around the bottom of the slot.
Experimental tests of the theory were performed, using 0.5–6 MHz ultrasonic pulses on duralumin cylinders with surface-breaking slots ranging from 0.5–2 mm in width and from 2–6 mm in depth. The numerical results were confirmed by these experiments.     

Traveltimes for infrasonic waves propagating in a stratified atmosphere

The tau– p method of Buland & Chapman (1983) is reformulated for sound waves propagating in a stratified atmosphere under the influence of a height-dependent wind velocity profile. For a given launch angle along a specified azimuth, the ray parameter is redefined to include the influence of the horizontal wind component along the direction of wave propagation. Under the assumption of negligible horizontal wind shear, the horizontal wind component transverse to the ray propagation does not affect the direction of the wave normal, but displaces the reference frame of the moving wavefront, thus altering the observed incidence azimuth. Expressions are derived for the time, horizontal range, and transverse range of the arriving waves as a function of ray parameter. Algorithms for the location of infrasonic wave sources using the modified tau– p formulation in conjunction with regional atmospheric wind and temperature data are discussed.     

Acoustic-Gravity Wave Generation at 120 km Altitude by Sea Level Detonation: A Preliminary Analysis of the Greene-Whitaker Calculation

Summary The Greene and Whitaker mechanism for the production of an acoustic- gravity wave in the 100-km region of the ionosphere is discussed from a physical point of view. The shock wave from a low altitude detonation propagates upward and, interacting with the atmosphere in the 100-km region, produces a horizontally propagating acoustic-gravity wave. Several aspects of this phenomenon are discussed. The initial period of the generated waves is in the 200–300 s range, the energy associated with the wave is about 0.3 per cent of the yield, and the wave generated appears to result principally from the refraction of the shock at the 100-km level.     

Energy flux in viscoelastic anisotropic media

We study properties of the energy-flux vector and other related energy quantities of homogeneous and inhomogeneous time-harmonic P and S plane waves, propagating in unbounded viscoelastic anisotropic media, both analytically and numerically. We propose an algorithm for the computation of the energy-flux vector, which can be used for media of unrestricted anisotropy and viscoelasticity, and for arbitrary homogeneous or inhomogeneous plane waves. Basic part of the algorithm is determination of the slowness vector of a homogeneous or inhomogeneous wave, which satisfies certain constraints following from the equation of motion. Approaches for determination of a slowness vector commonly used in viscoelastic isotropic media are usually difficult to use in viscoelastic anisotropic media. Sometimes they may even lead to non-physical solutions. To avoid these problems, we use the so-called mixed specification of the slowness vector, which requires, in a general case, solution of a complex-valued algebraic equation of the sixth degree. For simpler cases, as for SH waves propagating in symmetry planes, the algorithm yields simple analytic solutions. Once the slowness vector is known, determination of energy flux and of other energy quantities is easy. We present numerical examples illustrating the behaviour of the energy-flux vector and other energy quantities, for homogeneous and inhomogeneous plane P , SV and SH waves.     

Reply to comment by J. L. Mateos, O. Novaro, T. H. Seligman and J. Flores on 'Can horizontal P waves be trapped and resonate in a shallow sedimentary basin?'

Using a complete mathematical formulation, we show that the trapping of horizontal P waves in a very soft shallow alluvial layer is a minor effect. These waves do not have a stable way of propagation since in order to exist they require an incident wave and are therefore incapable of resonating in the lateral direction when confined in a basin of limited extent.     

Seismic body waves in anisotropic media: propagation through a layer

Summary. The square-root energy ratios and pulse shapes are presented for P, SV and SH waves transmitted through a layer of orthorhombic olivine between two isotropic half-spaces. Off incident planes of symmetry, incident P waves generate two small amplitude SH waves (one from each interface), whose amplitudes decrease slowly with increasing period. Incident SV (or SH ) waves can generate large amplitude SH (or SV ) waves which decrease rapidly with increasing period. For incident S waves, many pulses not present in isotropic models are generated, often of large relative amplitude, with many of the transmitted S pulses showing evidence of double arrivals, either in the form of S-wave splitting, or a modification of the shape of the input waveform.     

Existence of transverse waves in anisotropic poroelastic media

Out of the four waves in an anisotropic poroelastic medium, two are termed as quasi-transverse waves. The prefix 'quasi' refers to their polarizations being nearly, but not exactly, perpendicular to direction of propagation. In this composite medium, unlike perfectly elastic medium, the propagation of a longitudinal wave along a phase direction may not be accompanied by transverse waves. The existence of a transverse wave in anisotropic poroelastic media is ensured by the two equations restricting the choice of elastic coefficients of porous aggregate as well as fluid–solid coupling. Necessary and sufficient conditions for the existence of transverse waves along the coordinate axes and in the coordinate planes for general anisotropy are discussed. The discussion is extended to the case of orthotropic materials and existence for few specific phase directions is also explored. The conditions for the transverse waves decided on the basis of their apparent polarizations, that is, particle motion being perpendicular to ray direction, are also discussed. For a particular numerical model, the existence of these apparent transverse waves is solved numerically for phase directions in coordinate planes. For general directions of phase propagation, the existence of these transverse waves is checked graphically for the chosen numerical model.     

Asymptotic results for elastodynamic propagator matrices in plane-stratified and spherically-stratified earth models

Summary. Asymptotic results, applicable at high frequencies, are established for elastodynamic propagator or transfer matrices. We are concerned here with a single layer, by which we shall mean a region of a spherically-symmetric earth model in which physical properties vary smoothly with radius, or a region of a plane-stratified earth model in which physical properties vary smoothly with depth. Results are obtained for the spheroidal and toroidal oscillations of a solid spherical layer, the spheroidal oscillations of a spherical fluid layer, P-SV and SH waves in a plane solid layer and P waves in a plane fluid layer. Each of these results is given to first order in ω⁻¹ where ω is the angular frequency.     

Real and complex spectra – a generalization of WKBJ seismograms

Real plane-waves constitute the building blocks for recently developed spectral techniques in synthetic seismology. While providing numerical convenience, real slowness-spectra model certain wave phenomena in a distributed 'unnatural' way, whereas complex spectra model these phenomena in a compact, more 'natural' way. The theory of complex spectra, called by us the 'Spectral Theory of Transients' (STT) and developed elsewhere, is summarized here and contrasted with the real-spectrum approach. Relying strongly on the theory of analytic functions, STT permits the transient responses to be classified and evaluated according to the singularities they introduce in the complex slowness plane. The method is illustrated for a number of 2-D SH -wave model propagation environments, including interface reflection, head waves, multiple encounters with caustics due to concave boundaries or ducting medium inhomogeneities, and diffraction by structures with edges.