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.     

The variation of delays in stress-induced anisotropic polarization anomalies

Summary. Polarization anomalies in seismic shear wavetrains, diagnostic of propagation through anisotropic media, have now been observed in dilatancy zones in seismic regions. Stress-induced dilatancy will open cracks with preferred orientations, which will be effectively anisotropic to short-period seismic waves. The polarization anomalies are due to the shear waves splitting, in propagation through anisotropic media, into components with different polarizations and different velocities. This writes characteristic signatures into the shear wavetrains. The paper examines ways in which the differential shear-wave anisotropy (the delay between the split shear-waves) varies with direction by plotting stereograms of the relative delays, and their polarizations, for possible dilatancy symmetry-systems. It seems likely, that if sufficient observations of these anomalies can be obtained at each stage of the dilatancy episode, it will be possible to estimate the symmetry directions of the dilatancy and the geometry of the stress-induced crack-system, as well as monitoring the progress of the dilatancy episode.     

Shear-wave polarizations near the North Anatolian Fault – I. Evidence for anisotropy-induced shear-wave splitting

Summary. The Turkish Dilatancy Projects (TDP1 in 1979 and TDP2 in 1980) recorded small earthquakes near the North Anatolian Fault with closely-spaced networks of three-component seismometers in order to investigate the possibility of diagnosing dilatancy from its effects of shear-wave propagation. This paper examines the polarizations of shear wavetrains recorded in the shear-wave window immediately above the earthquake foci. Abrupt changes in the orientation and/or ellipticity of the shear-wave polarizations are almost always observed during the first few cycles following the initial shear-wave arrival on each seismogram. The horizontal projections of the polarizations of the first shear-wave arrivals at any given station show nearly parallel alignments with approximately the same orientations at each of the recording sites (with one exception). It is difficult to explain this uniform alignment over a wide area in terms of scattering at the irregular surface topography or by earthquake focal mechanisms. We demonstrate that the shear-wave splitting is likely to be the result of anisotropy in the region above the earthquake foci, which could produce polarizations displaying the observed alignments. The temporal change of the azimuth of alignment, observed at one locality between 1979 and 1980, may be due to the release of a local stress anomaly by a very near earthquake.     

Implementation of perfectly matched layers in an arbitrary geometrical boundary for elastic wave modelling

The perfectly matched layer (PML) absorbing boundary condition is incorporated into an irregular-grid elastic-wave modelling scheme, thus resulting in an irregular-grid PML method. We develop the irregular-grid PML method using the local coordinate system based PML splitting equations and integral formulation of the PML equations. The irregular-grid PML method is implemented under a discretization of triangular grid cells, which has the ability to absorb incident waves in arbitrary directions. This allows the PML absorbing layer to be imposed along arbitrary geometrical boundaries. As a result, the computational domain can be constructed with smaller nodes, for instance, to represent the 2-D half-space by a semi-circle rather than a rectangle. By using a smooth artificial boundary, the irregular-grid PML method can also avoid the special treatments to the corners, which lead to complex computer implementations in the conventional PML method. We implement the irregular-grid PML method in both 2-D elastic isotropic and anisotropic media. The numerical simulations of a VTI lamb's problem, wave propagation in an isotropic elastic medium with curved surface and in a TTI medium demonstrate the good behaviour of the irregular-grid PML method.     

B-polarization induction in two generalized thin sheets at the surface of a conducting half-space

Summary. A new closed-form solution is obtained analytically for a B- polarization induction problem of geophysical interest, in which a local region of the Earth is represented by a generalized thin sheet at the surface of and in electrical contact with a uniformly conducting half-space. The generalized sheet, first introduced by Ranganayaki & Madden, is a mathematical idealization of a double layer which consists, in this problem, of two adjacent half-planes with distinct conductances representing a surface conductivity discontinuity such as an ocean—coast boundary, underlain by a uniform sheet of finite integrated resistivity representing the lower crust. The resistive sheet exerts a considerable mathematical influence on the solution causing, under certain conditions, an additional pole to appear in one of the forms of contour integral by which the solution can be expressed; it also weakens or eliminates field singularities that would otherwise occur at the conductance discontinuity. A numerical calculation is made for model parameters typifying an ocean—coast boundary underlain by a highly resistive crust. It is found that the residue of the pole associated with the resistive sheet dominates the solution for this example, the main consequence of which is a huge increase in the horizontal range over which the induced currents adjust themselves between the different 'skin-effect' distributions at infinity on either side of the model. Moreover the solution shows that this 'adjustment distance' has a more complicated dependence on the conductance and integrated resistivity of the sheet than that given simply by the square root of their product which was the length parameter proposed by Ranganayaki & Madden.     

A simple method of representing azimuthal anisotropy on a sphere

We describe a method of expressing azimuthally anisotropic surface wave velocities on the Earth using a local and smooth spherical-spline parametrization. Anisotropy in the Earth leads to azimuthally varying Love and Rayleigh wave velocities that can be expressed as (cos 2ζ, sin 2ζ) and (cos 4ζ, sin 4ζ) perturbations to the isotropic velocities, where ζ is the direction of surface-wave propagation. The strength of the perturbations varies laterally, and a current goal of seismic tomography is the detailed global mapping of these variations. Several parametrizations have previously been used to describe azimuthally varying velocities. The representation proposed here uses spherical splines and is designed to describe smooth variations in both the strength and geometry of azimuthal anisotropy. The method builds on a simple geometrical approximation for the local azimuth of propagation expressed at the defining spline knot points. It avoids the singularities at the poles that result when azimuthal variations are parametrized using traditional scalar spherical harmonics. Compared with a generalized spherical-harmonic expansion of the tensor fields that represent 2ζ and 4ζ azimuthal variations smoothly on a sphere, the new method offers the advantages of local geographical support and simplicity of implementation.     

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

Summary. An improved finite difference scheme has been used to simulate the propagation of a plane P -impulse in an elastic half space with a slot normal to its surface. Various angles of incidence and dimensions of slot are considered. The numerical results are presented in several visualizations; each emphasized a different type of wave and all representations help in understanding the scattered and diffracted wave pattern. Experiments were carried out using 0.5—6 MHz ultrasonic pulses on a duralumin semidisc with a surface-breaking slot and the results are compared with those given by the numerical models.
The scattered wavefield includes compressional and Rayleigh pulses whose amplitude increases at the front of the slot and decreases behind it, as the angle of incidence is reduced. A diffracted compressional pulse is generated with a semicircular wavefront centred at the mid-point of the bottom of the slot. Also, two elliptical eddies are excited at the lower corners. These ellipses propagate into the medium and eventually spread out to form arc-shaped shear pulses.
In the shadow zone, behind the slot, the two components of displacement show independent behaviour. The horizontal component decreases either with decreasing angle of incidence or as the slot is made deeper. For acute angles, a reduction of displacement amplitude of about 50 per cent is obtained when the depth of the slot is made a half pulse width. On the other hand there is no diminution of the vertical displacement behind the slot, and, near the upper right corner, it is even amplified.     

Transient electromagnetic fields about an infinitesimally long grounded horizontal electric dipole on the surface of a uniform half-space

Summary. Laplace and Bessel Transforms are used to solve for the transient behaviour of the electromagnetic fields after switching off a steady current in a grounded infinitesimal horizontal dipole on the surface of a uniformly conducting half-space. Simple analytic expressions, which are valid for times sufficiently long after the switch that displacement terms can be ignored, are obtained on the surface of the half-space for the electric field and the time derivative of the magnetic field. At the instant of switching an infinitesimally long image becomes established directly under the source dipole. It is the diffusion of this image which gives the vertical magnetic field and horizontal electric fields their transient behaviour. During the transient, there is also a decaying charge distribution on the surface.     

Surface-wave propagation in a cracked poroelastic half-space lying under a uniform layer of fluid

The effect of cracks on the elastic properties of an isotropic elastic solid is studied when the cracks are saturated with a soft fluid. A polynomial equation in effective Poisson's ratio is obtained, whose coefficients are functions of Poisson's ratio of the uncracked solid, crack density and saturating fluid parameter. Elastic and dynamical constants used in Blot's theory of wave propagation in poroelastic solids are modified for the introduction of cracks. The effects of cracks on the velocities of three types of waves are observed numerically. The frequency equation is derived for the propagation of Rayleigh-type surface waves in a saturated poroelastic half-space lying under a uniform layer of liquid. Dispersion curves for a particular model of oceanic crust containing cracks are plotted. The effects of variations in crack density and saturation on the phase and group velocity are also analysed.     

Three-dimensional induction in a non-uniform thin sheet at the surface of a uniformly conducting earth

Summary. A new method for solving problems in three-dimensional electromagnetic induction in which the Earth is represented by a uniformly conducting half-space overlain by a surface layer of variable conductance is presented. Unlike previous treatments of this type of problem the method does not require the fields to be separated into their normal and anomalous parts, nor is it necessary to assume that the anomalous region is surrounded by a uniform structure; the model may approach either an E- or a B -polarization configuration at infinity. The solution is expressed as a vector integral equation in the horizontal electric field at the surface. The kernel of the integral is a Green's tensor which is expressed in terms of elementary functions that are independent of the conductance. The method is applied to an illustrative model representing an island near a bent coastline which extends to infinity in perpendicular directions.     

Love waves excited by discontinuous propagation of a rupture front

Summary. Analytical results are presented for Love waves generated by sudden changes of the rate of advance of a curved rupture front in an inclined fault plane that is embedded in an elastic half-space. The boundary condition at the surface of the half-space approximates the presence of an overlying layer. The calculation consists of two parts. First, ray theory is used to calculate far-field approximations to the horizontally polarized wavefields which are emitted when the speed of the rupture front suddenly changes. These fields can be expressed as products of emission coefficients (which govern the angular dependence) and propagation terms. Secondly, a representation integral for the Love wave over a surface enclosing the rupture front is constructed, using the emitted signal and an appropriate Green's function. This integral is evaluated asymptotically. The resulting approximate Love-wave spectrum shows an explicit dependence on the nature of the rupture process, on the rupture-front and fault-plane geometry, and on the magnitude of a sudden change in the rate of advance of the rupture front.     

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.     

Reflection—refraction of general P-and type-I S-waves in elastic and anelastic solids

Summary. The reflection and refraction of general (homogeneous or inhomo-geneous) plane P and type-I S ( SV ) body waves incident on plane boundaries are considered for general linear viscoelastic solids. Reflection—refraction laws, physical characteristics of the waves, and the nature of critical angles are examined in detail at welded boundaries and a free surface. General visco-elasticity with no low-loss approximations predicts that contrasts in intrinsic absorption at boundaries give rise to inhomogeneous reflected and refracted waves with elliptical particle motions, velocities and maximum attenuations that vary with frequency and angle of incidence, energy propagation at speeds and directions different from phase propagation, phase propagation that in general is parallel to the boundary for at most one angle of incidence, and reflection—transmission coefficients dependent on energy flow due to wave interaction. None of these physical characteristics are predicted for waves incident on boundaries that respond instantaneously.     

Complexity, self-organisation and variation in behaviour in meandering rivers

River meanders are natural features on the surface of Earth that present some degree of regularity of form. They range from being highly dynamic to being stable under present conditions. Conventional theory is that meanders develop to an equilibrium form which is related to discharge and sediment load. Other research has demonstrated that many highly active meanders exhibit a continuous evolution over time and a non-linearity in rate of development. Ideas of autogenesis and of self-organised criticality as being an explanation of some meander changes have been proposed. In this paper data from rivers around the world are examined for further evidence of autogenic, self-organised or non-linear behaviour through analysis of change in sinuosity over time for reaches and change in individual bend form, particularly bend curvature and bend elongation. Some examples do exhibit trends of increasing sinuosity over time and a few show limits from which large decreases occur. Several case studies show non-linearity of behaviour and increasing complexity of form. Other case studies, however, do not exhibit such trends. Phase space plots are used to help uncover emergent behaviour but show a variety of patterns. The example of a reach in which multiple cut-offs occurred is analysed for mechanisms of self-organisation of the planform and in the pool-riffle pattern. Riffles are more closely spaced and also more transient in the more rapidly changing and higher sinuosity parts of the channel. Hypothetical trajectories of different meander behaviour, including for bedrock meanders, are plotted but the challenge remains to uncover the conditions for occurrence and for divergence of tendencies to stability and instability. Identification of attractors and phase space of behaviour of different meandering systems offer the potential for application to sustainable channel management.     

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.     

Some remarks about the degree-one deformation of the Earth

The degree-one deformation of the Earth (and the induced discrepancy between the figure centre and the mass centre of the Earth) is computed using a theoretical approach (Love numbers formalism) at short timescales (where the Earth has an elastic behaviour) as well as at long timescales (where the Earth has a viscoelastic or quasi-fluid behaviour). For a Maxwell model of rheology, the degree-one relaxation modes associated with the viscoelastic Love numbers have been investigated: the M_o mode does not exist and there is only one transition mode (instead of two) generated by a viscosity discontinuity.
The translations at each interface of the incompressible layers of the earth model [surface, 670 km depth discontinuity, core-mantle boundary (CMB) and inner-core boundary (ICB)] are computed. They are elastic with an order of magnitude of about 1 mm when the excitation source is the atmospheric continental loading or a magnetic pressure acting at the CMB. They are viscoelastic when the earth is submitted to Pleistocene deglaciation, with an order of magnitude of about 1 m. In a quasi-fluid approximation (Newtonian fluid) because of the mantle density heterogeneity their order of magnitude is about 100 m (except for the ICB, which is in quasi-hydrostatic equilibrium at this timescale).     

The Resolving Power of Gross Earth Data

A gross Earth datum is a single measurable number describing some property of the whole Earth, such as mass, moment of interia, or the frequency of oscillation of some identified elastic-gravitational normal mode. We show how to determine whether a given finite set of gross Earth data can be used to specify an Earth structure uniquely except for fine-scale detail; and how to determine the shortest length scale which the given data can resolve at any particular depth. We apply the general theory to the linear problem of finding the depth-variation of a frequency-independent local Q from the observed quality factors Q of a finite number of normal modes. We also apply the theory to the non-linear problem of finding density vs depth from the total mass, moment, and normal-mode frequencies, in case the compressional and shear velocities are known.     

Head waves in laterally heterogeneous media

Summary. A layer of constant thickness over a half-space is assumed, and the propagation of head waves is considered for the following two cases: (1) the P -wave velocity varies in the layer in the horizontal direction, and is constant in the half-space: (2) the P -wave velocity varies in the half-space in the horizontal direction, and is constant in the layer. In each case the horizontal velocity gradient is assumed to remain constant. The wave propagation is investigated in the direction of the gradient (direct profile), and opposite to it (reverse profile). Formulae for the travel times and the amplitudes are obtained on the basis of ray-theoretical considerations. Conditions are discussed for the discrimination in a field experiment between the case of a sloping boundary separating the homogeneous media, and the case of an intrinsic horizontal velocity gradient.     

Quasi-static deformation of a poroelastic half-space with anisotropic permeability by two-dimensional surface loads

An analytical solution is obtained of the fully coupled diffusion–deformation system of equations governing the quasi-static plane strain deformation of a poroelastic half-space with anisotropic permeability and compressible constituents. The stresses and the pore pressure are taken as the basic state variables. Displacements are obtained by integrating the coupled constitutive relations. The problem of surface loads is discussed in detail. Explicit analytical solutions are derived for normal line loading, shear line loading and normal strip loading. The permeability anisotropy is found to have a significant effect on the quasi-static deformation of the half-space. However, in the drained and undrained limits, the anisotropy has no effect. The stresses in the drained and undrained states are independent of the poroelastic parameters. Numerical computation of the pore pressure indicates that ignoring permeability anisotropy may lead to an overestimation of the pore pressure at points vertically below the point of normal loading. Further, anisotropy in permeability may lead to a dilution in the theoretical prediction of the Mandel–Cryer Effect.     

Regional tomographic inversion of the amplitude and phase of Rayleigh waves with 2-D sensitivity kernels

In this study, we test the adequacy of 2-D sensitivity kernels for fundamental-mode Rayleigh waves based on the single-scattering (Born) approximation to account for the effects of heterogeneous structure on the wavefield in a regional surface wave study. The calculated phase and amplitude data using the 2-D sensitivity kernels are compared to phase and amplitude data obtained from seismic waveforms synthesized by the pseudo-spectral method for plane Rayleigh waves propagating through heterogeneous structure. We find that the kernels can accurately predict the perturbation of the wavefield even when the size of anomaly is larger than one wavelength. The only exception is a systematic bias in the amplitude within the anomaly itself due to a site response.
An inversion method of surface wave tomography based on the sensitivity kernels is developed and applied to synthesized data obtained from a numerical simulation modelling Rayleigh wave propagation over checkerboard structure. By comparing recovered images to input structure, we illustrate that the method can almost completely recover anomalies within an array of stations when the size of the anomalies is larger than or close to one wavelength of the surface waves. Surface wave amplitude contains important information about Earth structure and should be inverted together with phase data in surface wave tomography.