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.     

Seismic waves in a stratified half space

Summary. For a buried source in a stratified elastic half space, the surface displacements are calculated by numerical integration of the Fourier–Bessel transform of the response. In the transform space this response is conveniently represented in terms of the reflection and transmission properties of the half space. For a layered medium this procedure avoids all problems associated with growing exponential terms in the evanescent regime. A slightly attenuative medium is assumed, so that the surface wave poles are shifted off the real slowness axis and thus a contour of integration along this axis may be employed. A general point source is represented by an arbitrary moment tensor.
The procedure is illustrated by calculations of three component seismograms including all P , SV and SH contributions for body and surface waves at moderate ranges. For local earthquakes we illustrate the striking effect of focal depth and also show the effect of sedimentary cover on strong ground motion.     

Elastic wave propagation in model sediments – II

Summary. A fluid-saturated cubic packing of like elastic spheres is taken to be in equilibrium under the effect of gravity and the effects of a superimposed low-frequency elastic wave are considered. In the first place, expressions for the wave velocity, dispersion and attenuation are derived for the dry packing. This dynamic theory leads to the result that, for very low frequencies, the wave velocity is proportional to the third root of the depth and not the sixth root as is obtained by using the effective elastostatic modulus of the packing. For the fluid-saturated packing, two waves, termed respectively the 'solid wave' and the 'fluid wave', are found to propagate. The 'solid wave' has the characteristics of a wave propagating within a dry packing whose parameters differ in a specified way from those of the original packing, whereas the 'fluid wave' has those of a wave within a homogeneous fluid with similarly modified parameters.     

Full reconstruction of a layered elastic medium from P-SV slant-stack data

Summary. We develop a méthod of reconstructing the elastic paraméters as functions of depth, for a horizontally stratified, isotropic elastic half-space. Unlike previous schemes, which have been able to retrieve the shear wave speed and density from SH seismograms slant stacked at two angles, our méthod makes use of P - SV data at a single stacking paraméter to obtain all three elastic constants. The data required are the elements of the full reflection matrix at the surface, corresponding to measurements of two separate components of the response to two independent sources, one explosive, the other generating shear waves.
In developing this inverse scheme fundamental differences emerge between the acoustic or SH problem, and the coupled P - SV case, the most important being in the nature of the interfacial scattering matrix. We show that it is not possible to make use of the downward reflection data for an interface to determine directly the remaining reflection and transmission coefficients, but that the scattering data may be completed by applying a simple iterative procedure at each interface.
We show the result of applying our inverse scheme to seismograms generated for a six-layered model, including a low-velocity layer. We are able to reconstruct both wave speeds and the density as functions of depth, all quantities being in close agreement with the original model.     

A Preliminary Investigation into the Relationship between Long-Period Seismic Noise and Local Fluctuations in the Atmospheric Pressure Field

Summary The displacement response of an elastic half space to a plane pressure wave is examined in order to establish the conditions under which sources of this type can contribute significantly to the long-period seismic noise field. The study is restricted to pressure waves which propagate at velocities well below the seismic wave velocities characteristic of the half space. The numerical studies indicate that pressure waves with amplitudes of 100 μbar or more can contribute significantly to the long-period vertical background noise observed at the surface, provided that the detectors are located on sections of alluvial fill or poorly to moderately indurated sandstones and shales whose thicknesses are greater than about a kilometre. These same waves can also create significant tilt noise on long-period horizontal seismographs located at or near the surface, regardless of the rock type. The seismic disturbances created by pressure waves decay rapidly away from the surface. Therefore, it appears that it may be possible to eliminate the effects of atmospherically generated noise by placing the detectors at moderate depths.     

On Love Waves across the Ocean

i
Displacements of Love waves generated by a two-dimensional point source in a layered medium have been studied earlier by Sezawa & Sato by the method of successive reflections at the boundaries. In this paper the same problem has been worked out by using Green's function. The paper deals with the study of attenuation of Love waves of low periods in the coastal region. Experimental observations show that Love waves of smaller periods can be obtained only in the island observing stations. A slight intervention of the continental boundary is sufficient to attenuate lower period Love waves giving a hint thereby that attenuation of lower periods takes place perhaps at the continental margin. Taking a simplified configuration for the continental boundary and using Green's function technique, the displacement of Love waves due to a point source has been obtained and it has been shown that attenuation of Love waves of smaller periods takes place in the continental margin due to the slope of the boundary.     

Guided wave propagation in laterally varying media - I. Theoretical development

Summary. The propagation of surface waves in a laterally varying medium can be described by representing the wavetrain as a superposition of modal contributions for a reference structure. As the guided waves propagate through a heterogeneous zone the modal coefficients needed to describe the wavetrain vary with position, leading to interconversions between modes and reflection into backward travelling modes. The evolution of the modal terms may be described by a set of first-order differential equations which allow for coupling to both forward and backward travelling waves; the coefficients in these equations depend on the differences between the actual structure and the reference structure. This system is established using the orthogonality properties of the modal eigenfunctions and is valid for SH -waves, P - SV -waves and full anisotropy.
The reflected and transmitted wavefields for a region of heterogeneity can be related to the incident wave by introducing reflection and transmission matrices which connect the modal coefficients in these fields to those in the incident wavetrain. By considering a sequence of models with increasing width of heterogeneity we are able to derive a set of Ricatti equations for the reflection and transmission matrices which may be solved by initial value techniques. This avoids an awkward two-point boundary value problem for a large number of coupled equations. The method is demonstrated for 1 Hz Lg - and Sn -waves in a multilayered model for which there are 19 coupled modes.
The method is applicable to three-dimensional heterogeneity, and we are able to show that the interconversion between Love and Rayleigh waves, in the presence of gradients in seismic properties transverse to the propagation path, leads to a net rate of increase of the transverse components of the seismogram at the expense of the other components.     

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.     

A review of the effects of anisotropic layering on the propagation of seismic waves

Summary. This paper reviews recent work, much of it unpublished, on the effects of anisotropy on seismic waves, and lays the theoretical background for some of the other papers in this number of the Geophysical Journal .
The propagation of both body and surface waves in anisotropic media is fundamentally different from their propagation in isotropic media, although the differences in behaviour may be comparatively subtle and difficult to observe. One of the most diagnostic of these anomalies, which has been observed on some surface-wave trains, and should be evident in body-wave arrivals, is generalized, three-dimensional polarization, where the Rayleigh motion is coupled to the Love, and the P and SV motion is coupled to the SH . This coupling introduces polarization anomalies which may be used to investigate anisotropy within the Earth.     

Point Sources of Elastic Deformation: Elementary Sources, Static Displacements

Summary. We construct a catalogue of all possible elementary point sources for static deformation in an elastic solid. the familiar double-couples, CLVD's, centres of compression and dilatation, etc., are all members of the complete catalogue. the sources are classified according to the rank of the seismic moment tensor, and according to the weight (or order) of the irreducible tensor representation of the 3-D rotation group. These sources can be classified as belonging to one of three general classes. the static excitation functions are calculated for an infinite, homogeneous, isotropic medium for all these sources. We show that, except for sources belonging to these three general classes, all other sources — which are numerous for the tensors of high rank — are null static sources. That is, sources that do not produce any static displacement outside of the source region. Due to the presence of null sources, an inversion of the static deformation data is non-unique. the expansion of the equivalent-force tensors and the stress glut tensors (or seismic moment tensors) into a set of the symmetric trace-free source tensors is proposed. the sources corresponding to seismic moment tensors of the second, third and fourth ranks are considered in more detail. We identify the third-rank sources with rotational dislocations or disclinations.     

One-way representations of seismic data

A general one-way representation of seismic data can be obtained by substituting a Green's one-way wavefield matrix into a reciprocity theorem of the convolution type for one-way wavefields. From this general one-way representation, several special cases can be derived.
By introducing a Green's one-way wavefield matrix for primaries , a generalized Bremmer series representation is obtained. Terminating this series after the first-order term yields a primary representation of seismic reflection data. According to this representation, primary seismic reflection data are proportional to a reflection operator, 'modified' by primary propagators for downgoing and upgoing waves. For seismic imaging, these propagators need to be inverted. Stable inverse primary propagators can easily be obtained from a one-way reciprocity theorem of the correlation type.
By introducing a Green's one-way wavefield matrix for generalized primaries , an alternative representation is obtained in which multiple scattering is organized quite differently (in comparison with the generalized Bremmer series representation). According to the generalized primary representation, full seismic reflection data are proportional to a reflection operator, 'modified' by generalized primary propagators for downgoing and upgoing waves. Internal multiple scattering is fully included in the generalized primary propagators {either via a series expansion or in a parametrized way). Stable inverse generalized primary propagators can be obtained from the one-way reciprocity theorem of the correlation type. These inverse propagators are the nucleus for seismic imaging techniques that take the angle-dependent dispersion effects due to fine-layering into account.     

What can be learned from rotational motions excited by earthquakes?

One answer to the question posed in the title is that we will have more accurate data for arrival times of SH waves, because the rotational component around the vertical axis is sensitive to SH waves although not to P-SV waves. Importantly, there is another answer related to seismic sources, which will be discussed in this paper.
Generally, not only dislocations commonly used in earthquake models but also other kind of defects could contribute to producing seismic waves. In particular, rotational strains at earthquake sources directly generate rotational components in seismic waves. Employing the geometrical theory of defects, we obtain a general expression for the rotational motion of seismic waves as a function of the parameters of source defects.
Using this expression, together with one for translational motion, we can estimate the rotational strain tensor and the spatial variation of slip velocity in the source area of earthquakes. These quantities will be large at the edges of a fault plane due to spatially rapid changes of slip on the fault and/or a formation of tensile fractures.     

Elastodynamic and elastostatic Green tensors for homogeneous weak transversely isotropic media

An explicit analytical formula for the complete elastodynamic Green tensor for homogeneous unbounded weak transversely isotropic media is presented. The formula was derived by analytical calculations of higher-order approximations of the ray series. The ray series is finite and consists of seven non-zero terms. The formula for the Green tensor is complete and correct for the whole frequency range, thus it describes correctly the wavefield at all distances and at all directions including the shear-wave singularity direction. The Green tensor consists of P, SV and SH far-field waves and four coupling waves. Three of them couple P and SV waves, and the fourth wave couples the SV and SH waves. The P-SV coupling waves behave similarly to the near-field waves in isotropy. However, the SV-SH coupling wave, which is called 'shear-wave coupling', behaves exceptionally and it has no analogy in the Green tensor for isotropy. The formula for the elastostatic Green tensor is also derived.     

The effects of post-seismic motions on the moment of inertia of a stratified viscoelastic earth with an asthenosphere

Summary. We give the analytical formulation for calculating the transient displacement of fields produced by earthquakes in a stratified, selfgravitating, incompressible, viscoelastic earth. We have evaluated the potential of viscous creep in the asthenosphere in exciting the Chandler wobble by a four-layer model consisting of an elastic lithosphere, a two-layer Maxwell viscoelastic mantle, and an inviscid core. The seismic source is modelled as an inhomogeneous boundary condition, which involves a jump condition of the displacement fields across the fault in the lithosphere. The response fields are derived from the solution of a two-point boundary value problem, using analytical propagator matrices in the Laplace-transformed domain. Transient flows produced by post-seismic rebound are found to be confined within the asthenosphere for local viscosity values less than 10²⁰P. The viscosity of the mantle below the low-viscosity channel is kept at 10²²P. For low-viscosity zones with widths greater than about 100 km and asthenospheric viscosities less than 10¹⁸P, we find that viscoelasticity can amplify the perturbations in the moment of inertia by a factor of 4–5 above the elastic contribution within the time span of the wobble period. We have carried out a comparative study on the changes of the inertia tensor from forcings due to surface loading and to faulting. In general the global responses from faulting are found to be much more sensitive to the viscosity structure of the asthenosphere than those produced from surface loading.     

The elastodynamic near field

Summary. The elastodynamic fields of point forces and shear dislocations of finite source duration are analysed with the aim of establishing the frequency and time-domain characteristics of the field in the near-source region. Criteria are obtained for amplitude dominance in regions where the source–sensor distance is much smaller than the wavelength.
It is shown that in the frequency domain , the Green's tensor (and hence the displacement field of a single point force) attenuates like R ⁻¹ in the near-source region and there exists no region in which the 'near-field' term becomes dominant such that the 'far-field' term can be neglected. Hence, there is no real 'near-field' term for the elastodynamic Green's tensor. The near-field terms of the displacements, velocities and accelerations excited by a shear dislocation attenuate like R ⁻², since the R ⁻³ and R ⁻⁴ terms tend to be eliminated due to mutual cancellation of P and S motions in the near-source region.
In the time domain , the corresponding near field of the displacement field is defined for the steady amplitude interval (away from transients) R /β < t < R /α+ T by the condition R ≤βT where β is the shear velocity and T is the source's duration. The relative strengths of all other arrivals will depend on the particular time window under consideration.
The particle motion patterns due to a single force in the near-source region are shown to be similar to rotating hyperbolas with an axis along the force direction, which are quite different from the 'smoke ring' motion patterns of the so-called 'near-field' term itself.     

Centroid moment tensor inversion of low-frequency seismic spectra using Green's functions for aspherical earth models

We present a new method for centroid moment tensor (CMT) inversion, in which we employ the Green's function computed for aspherical earth models using the Direct Solution Method. We apply this method to CMT inversion of low-frequency seismic spectra for the 1994 Bolivia and 1996 Flores Sea deep earthquakes. The estimated centroid locations agree well with those obtained by multiple-shock analyses using body-wave data. This shows that it is possible to obtain reliable CMT solutions by analyses of low-frequency seismic spectra using accurate Green's functions computed for present 3-D earth models.     

Numerical simulation of the propagation of P waves in fractured media

We study the propagation of P waves through media containing open fractures by performing numerical simulations. The important parameter in such problems is the ratio between crack length and incident wavelength. When the wavelength of the incident wavefield is close to or shorter than the crack length, the scattered waves are efficiently excited and the attenuation of the primary waves can be observed on synthetic seismograms. On the other hand, when the incident wavelength is greater than the crack length, we can simulate the anisotropic behaviour of fractured media resulting from the scattering of seismic waves by the cracks through the time delay of the arrival of the transmitted wave. The method of calculation used is a boundary element method in which the Green's functions are computed by the discrete wavenumber method. For simplicity, the 2-D elastodynamic diffraction problem is considered. The rock matrix is supposed to be elastic, isotropic and homogeneous, while the cracks are all empty and have the same length and strike direction. An iterative method of calculation of the diffracted wavefield is developed in the case where a large number of cracks are present in order to reduce the computation time. The attenuation factor Q ⁻¹ of the direct waves passing through a fractured zone is measured in several frequency bands. We observe that the attenuation factor Q ⁻¹ of the direct P wave peaks around kd = 2, where k is the incident wavenumber and d the crack length, and decreases proportionally to ( kd ) ⁻¹ in the high-wavenumber range. In the long-wavelength domain, the velocity of the direct P wave measured for two different crack realizations is very close to the value predicted by Hudson's theory on the overall elastic properties of fractured materials.     

Seismic waves in rocks with fluids and fractures

Seismic wave propagation through the earth is often strongly affected by the presence of fractures. When these fractures are filled with fluids (oil, gas, water, CO₂, etc.), the type and state of the fluid (liquid or gas) can make a large difference in the response of the seismic waves. This paper summarizes recent work on methods of deconstructing the effects of fractures, and any fluids within these fractures, on seismic wave propagation as observed in reflection seismic data. One method explored here is Thomsen's weak anisotropy approximation for wave moveout (since fractures often induce elastic anisotropy due to non-uniform crack-orientation statistics). Another method makes use of some very convenient crack/fracture parameters introduced previously that permit a relatively simple deconstruction of the elastic and wave propagation behaviour in terms of a small number of crack-influence parameters (whenever this is appropriate, as is certainly the case for small crack densities). Then, the quantitative effects of fluids on these crack-influence parameters are shown to be directly related to Skempton's coefficient B of undrained poroelasticity (where B typically ranges from 0 to 1). In particular, the rigorous result obtained for the low crack density limit is that the crack-influence parameters are multiplied by a factor  (1 − B )  for undrained systems. It is also shown how fracture anisotropy affects Rayleigh wave speed, and how measured Rayleigh wave speeds can be used to infer shear wave speed of the fractured medium in some cases. Higher crack density results are also presented by incorporating recent simulation data on such cracked systems.     

Symmetries in the reflection and transmission of elastic waves

Summary. The symmetry relations between the reflection and transmission coefficients for plane elastic waves incident upon an arbitrary horizontally stratified medium are derived by a novel approach. Previous results, particularly for a single interface, are obtained as special cases of this treatment.
In addition, for perfectly elastic media, projection operators for travelling and evanescent waves are introduced and used to derive a number of new relationships between the reflection and transmission coefficients.