Cagniard de-Hoop method for a Haskell type vertical fault

Summary. There are two ways to apply the Cagniard de-Hoop method when generating synthetic seismograms due to a source in a three-dimensional medium. One is the Hankel transform method (hereafter called 'the cylindrical wave representation method'), which utilizes the property of the modified Bessel function. The other is 'the plane wave representation method', which replaces the Bessel function by a superposition of plane waves. In extending a point source to a finite dimensional source, the latter method is extremely useful, because it enables the integrations on the fault surface to be performed analytically.
Using the latter method, expressions for displacements due to a Haskell type vertical fault in a uniform half-space are obtained. A solution is given as a sum of four quadrantal source contributions, which is similar to Madariaga's solution for a source in the whole space. Each contribution consists of a single finite range integration or a single integration plus a pole contribution, depending on the location of the observation point with respect to the source. The procedure can be extended to other fault types.     

Seismic waves in a stratified half space

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.     

Seismic waves in stratified anisotropic 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 symmétry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmétry; 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.     

Flexure of the ocean lithosphere from island uplift, bathymetry and geoid height observations: the Society Islands

Summary. The response of a stratified elastic medium can be conveniently characterized by the Green's tensor for the medium. For coupled seismic wave propagation ( P—SV or fully anisotropic), the Green's tensor may be constructed directly from two matrices of linearly independent displacement solutions. Rather simple forms for the Green's tensor can be found if each displacement matrix satisfies one of the boundary conditions on the seismic field. This approach relates directly to 'reflection matrix' representations of the seismic field.
For a stratified elastic half space the Green's tensor is used to give a spectral representation for coupled seismic waves. By means of a contour integration a general completeness relation is obtained for the 'body wave' and 'surface wave' parts of the seismic field. This relation is appropriate for SH and P–SV waves in an isotropic medium and also for full anisotropy.     

Radiation from an impulsive line source in an unbounded homogeneous anisotropic medium

Summary. The space-time elastic wave motion generated by an impulsive line source in a homogeneous anisotropic medium is calculated with the aid of the Cagniard-de Hoop method. Two types of sources are considered in detail, viz. a line source of expansion (model for an explosive source) and a line force (model for a mechanical vibrator). Numerical results are presented for the radiated particle velocity in the medium including those regions of space where shear-wave triplication occurs. There is a marked difference in the time response observed for the two types of sources and for the different positions of the receiver with respect to the source position. These waveform differences are important when the radiated wave is used to determine experimentally the elastic properties of the medium. As compared with the traditional Fourier-integral transform method to handle this problem, the computation time with the present method is considerably less.     

P-wave seismograms from continuously varying transition zones—II. Elastic media

Summary. A class of elastic transition zones are modelled by considering a homogeneous half space overlying an inhomogeneous half space with a bounded and monotonically increasing profile for the rigidity modulus and constant Poisson's ratio and density. Reflected P waves due to a compressional point source in the upper half space are studied in the frequency and time domains by means of numerical contour integration in the complex k plane and the Fast Fourier Transform (FFT). Results from the exact fourth-order elasticity theory are compared with those from the approximate decoupled equations for P and SV waves. Agreement is observed between the two theories at high frequencies beyond the caustic range.     

Automatic model for finding the one-dimensional magnetotelluric problem

Summary. A method is described for finding a resistivity model that fits given magnetotelluric data in the one-dimensional case. The procedure is automatic and objective in that no a priori model structure is imposed. Starting with a uniform half space derived directly from the data, the procedure gradually transforms the half space to one with a continuous and smooth resistivity distribution whose response fits the measured data. The method is illustrated by application to two magnetotelluric data sets.     

Fingerprinting the Origin of Fluvial Suspended Sediment in Larger River Basins: Combining Assessment of Spatial Provenance and Source Type

Investigation of the potential for using sediment fingerprinting to integrate both spatial provenance and source type information for larger drainage basins appears to be desirable. This contribution presents the results of adopting a composite fingerprinting procedure incorporating statistically verified multicomponent signatures and a multivariate mixing model to provide a preliminary integration of spatial provenance and source type information for the upper and middle reaches of the drainage basins of the Rivers Exe (601 km²) and Severn (4325 km²), UK. A nested approach is employed, whereby spatial provenance is addressed in terms of the distinct sub-basin zones constituting each study area as an entirety, and source type is then characterised within each of these distinct spatial zones in terms of surface (woodland, pasture, cultivated) and subsurface (channel bank) materials. The results demonstrate that the fingerprinting approach possesses considerable potential for integrating spatial provenance and source type information, and hence for improving the resolution of existing sediment source information for larger drainage basins.     

Kirchhoff synthetics for seismic reflections and diving waves in two dimensions and application to the D" layer

The Kirchhoff-Helmholtz (KH) integration has been used to model the reflected and the diving waves from an interface with a positive velocity gradient. The modelling is carried out for a spherical boundary and for a sinusoidal topography with a long-scale wavelength.
An artefact, which is a major problem in modelling the seismic response using the KH integration, has been reduced by introducing a Hilbert transform sign manipulation. Cleaner synthetic seismograms with correct amplitudes have been produced by this method. A discretization in larger surface elements has been made possible by introducing a smoothing factor that suppresses the noise that normally follows the constructed signal if a large element size is taken.     

A WKBJ spectral method for computation of SV synthetic seismograms in a cylindrically symmetric medium

Summary. A method of synthetic seismogram computation for teleseismic SV -waves is developed in order to treat quantitatively SV -waves in problems of body wave source inversion and source—receiver structure studies. The method employs WKBJ theory for a generalized ray in a vertically inhomogeneous half-space and the propagator matrix technique for waves in near-surface homogeneous layers. Wavenumber integration is done along the real axis of the wavenumber plane and anelasticity is included by using complex velocity in all regions of the earth model. The near-surface source structure is taken into account in the computation for the case of the shallow source by allowing a point source to be located in the homogeneous layers. Source and receiver area structures are also allowed to differ. A general moment tensor point source is considered.     

Volatility of Bitumen Prices and Implications for the Industry

Sustained crude oil price increases have led to increased investment in and production of Canadian bitumen to supplement North American oil supplies. For new projects, the evaluation of profitability is based on a prediction of the future price path of bitumen and ultimately light/medium crude oil. This article examines the relationship between the bitumen and light crude oil prices in the context of a simple error-correction economic-adjustment model. The analysis shows bitumen prices to be significantly more volatile than light crude prices. Also, the dominant effect of an oil price shock on bitumen prices is immediate and is amplified, both in absolute terms and percentage price changes. It is argued that the bitumen industry response to such market risks will likely be a realignment toward vertical integration via new downstream construction, mergers, or on a de facto basis by the establishment of alliances.     

Slowness—time mapping of near offset seismic reflection data

Summary. The transformation of a set of seismograms to the delay time-slowness, τ—p, domain is presented as a sequence of Fourier and Bessel transforms, For a horizontally layered medium, this sequence gives an exact cylindrical wave decomposition of the response to a point source; correctly compensating for the phase shifting and geometrical spreading associated with transmission through the Earth. The resultant τ—p map or 'slant stack' contains true amplitude and phase information. The spatial aliasing properties of the transformation, when applied to a dataset, are greatly improved by the use of only outgoing waves in the Bessel transform. This is equivalent to using Hankel functions rather than Bessel functions, and is justified by the absence of incoming waves from most datasets. The WKBJ approximation to the medium response enables predictions to be made about the shape and amplitude variation with slowness of truncation effects. Theoretically the τ—p transformation is reversible, thus the τ—p domain is a suitable one in which to perform filtering operations before seismogram reconstruction.     

On the frequency contents of local events: source or path effect?

Local events which occurred in the Timanfaya volcanic field are analysed and their spectral contents are interpreted as being due to the effect of a finely layered medium with large impedance contrast, the presence of which is well known from well-logging data. By considering the layered medium as a filter, its spectral response (amplitude and phase) is computed from different randomly generated values of its parameters (total thickness, mean thickness of the individual layers and impedance contrast), showing that this stratigraphic filter acts as a low-pass filter. This effect can be correctly modelled by means of an exponential correlation function. However, in some cases resonant peaks appear in the spectrum superimposed on the low-pass filter effect, which can only be attributed to resonant scattering. An analysis of the phase and group delay associated with the amplitudes is made and strongly suggests that it is necessary to incorporate the theory of resonant scattering for the retrieval of medium properties. From the point of view of the amplitudes, the superposition of both effects, low-pass filter and resonant peaks, allows us to reproduce the gross features that are usually attributed to source effects. The need for a careful analysis of the local site response previous to any interpretation of the spectral features in terms of source characteristics is thus emphasized.     

P-SH conversions in a flat-layered medium with anisotropy of arbitrary orientation

P-SH conversion is commonly observed in teleseismic P waves, and is often attributed to dipping interfaces beneath the receiver. Our modelling suggests an alternative explanation in terms of flat-layered anisotropy. We use reflectivity techniques to compute three-component synthetic seismograms in a 1-D anisotropic layered medium. For each layer of the medium, we prescribe values of seismic velocities and hexagonally symmetric anisotropy about a common symmetry axis of arbitrary orientation. A compressional wave in an anisotropic velocity structure suffers conversion to both SV -and SH -polarized shear waves, unless the axis of symmetry is everywhere vertical or the wave travels parallel to all symmetry axes. The P-SV conversion forms the basis of the widely used 'receiver function' technique. The P-SH conversion occurs at interfaces where one or both layers are anisotropic. A tilted axis of symmetry and a dipping interface in isotropic media produce similar amplitudes of both direct ( P ) and converted ( Ps ) phases, leaving the backazimuth variation of the P-Ps delay as the main discriminant. Seismic anisotropy with a tilted symmetry axis leads to complex synthetic seismograms in velocity models composed of just a few flat homogeneous layers. It is possible therefore to model observations of P coda with prominent transverse components with relatively simple 1-D velocity structures. Successful retrieval of salient model characteristics appears possible using multiple realizations of a genetic-algorithm (GA) inversion of P coda from several backazimuths. Using GA inversion, we determine that six P coda recorded at station ARU in central Russia are consistent with models that possess strong (> 10 per cent) anisotropy in the top 5 km and between 30 and 43 km depth. The symmetry axes are tilted, and appear aligned with the seismic anisotropy orientation in the mantle under ARU suggested by SKS splitting.     

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.     

Surface wave tomography for azimuthal anisotropy in a strongly reduced parameter space

Large scale seismic anisotropy in the Earth's mantle is likely dynamically supported by the mantle's deformation; therefore, tomographic imaging of 3-D anisotropic mantle seismic velocity structure is an important tool to understand the dynamics of the mantle. While many previous studies have focused on special cases of symmetry of the elastic properties, it would be desirable for evaluation of dynamic models to allow more general axis orientation. In this study, we derive 3-D finite-frequency surface wave sensitivity kernels based on the Born approximation using a general expression for a hexagonal medium with an arbitrarily oriented symmetry axis. This results in kernels for two isotropic elastic coefficients, three coefficients that define the strength of anisotropy, and two angles that define the symmetry axis. The particular parametrization is chosen to allow for a physically meaningful method for reducing the number of parameters considered in an inversion, while allowing for straightforward integration with existing approaches for modelling body wave splitting intensity measurements. Example kernels calculated with this method reveal physical interpretations of how surface waveforms are affected by 3-D velocity perturbations, while also demonstrating the non-linearity of the problem as a function of symmetry axis orientation. The expressions are numerically validated using the spectral element method. While challenges remain in determining the best inversion scheme to appropriately handle the non-linearity, the approach derived here has great promise in allowing large scale models with resolution of both the strength and orientation of anisotropy.     

Extension of the Thomson-Haskell method to non-homogeneous spherical layers

Summary. Amplitude spectra of Rayleigh and Love waves in a layered non-gravitating spherical earth have been obtained using as a source, displacement and stress discontinuities. In each layer elastic parameters and density follow specified functions of radial distance and the solutions of the equations of motion are obtained in terms of exponential functions. The Thomson—Haskell method is extended to this case. The problem reduces to simple calculations as in a plane-layered medium. Numerical results of phase and group velocities up to periods of 300 s in various earth models when compared with earlier results (obtained by numerical integration) show that the present method can be used with sufficient accuracy. The differences in phase velocity, group velocity and amplitude (also surface ellipticity in the case of Rayleigh waves) between spherical- and flat-earth models have been investigated in the range 20–300–s period and expressed in polynomials in the period.     

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.     

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.