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.     

Kirchhoff–Helmholtz reflection seismograms in a laterally inhomogeneous multi-layered elastic medium – II. Computations

Summary. High-frequency reflection and refraction seismograms for laterally variable multi-layered elastic media are computed by using the frequency domain elastic Kirchhoff–Helmholtz (KH) theory of Frazer and Sen. Both source and receiver wavefields are expanded in series of generalized rays and then elastic (KH) theory is applied to determine the coupling between each source ray and each receiver ray at each interface. The motion at the receiver is given as a series of integrals, one for each generalized ray. We use geometrical optics and plane wave reflection and transmission coefficients for rapid evaluation of the integrand. When the source or the receiver ray field has caustics on the surface of integration geometrical ray theory breaks down and this gives rise to singularities in the KH integrand. We repair this using methods suggested by Frazer and Sen.
Examples of reflection seismograms for 2-D structures computed by elastic KH theory are shown. Those for a vertical fault scarp structure are compared with the seismograms obtained by physical modelling. Then OBS data obtained from the mid-America trench offshore Guatemala area are analysed by computing KH synthetics for a velocity model that has been proposed for that area. Our analysis indicates the existence of a small low-velocity zone off the trench axis.
No head wave arrivals are obtained in our KH synthetics since we do not consider multiple interactions of a ray with an interface. The nearly discontinuous behaviour of elastic R/T coefficients near the critical angle causes small spurious phases which arrive later than the correct arrivals.     

Induction in a thin sheet of variable conductance at the surface of a stratified earth – II. Three-dimensional theory

Summary. The algorithm of Dawson & Weaver for modelling electromagnetic induction effects in a thin sheet at the surface of a uniform earth is modified to permit the use of a layered earth model. The theory is developed in Fourier space in terms of the toroidal and poloidal transfer functions instead of with the Green's function approach which was used by Dawson & Weaver. The integral equation for the surface electric field and most of the integral formulae for the derived field components are the same as before, except for the inclusion of additional integral the kernel of which has to be calculated numerically with the aid of fast Hankel transforms. The accuracy of the results is tested by comparing solutions with those obtained from a related 2-D algorithm and finally an example of 3-D modelling is presented.     

Transient and impulse responses of a one-dimensional linearly attenuating medium—II. A parametric study

Summary. We investigate one-dimensional waves in a standard linear solid for geophysically relevant ranges of the parameters. The critical parameters are shown to be T*= t_u/Q_m where t _u is the travel time and Q_m the quality factor in the absorption band, and τ⁻¹ _m , the high-frequency cut-off of the relaxation spectrum. The visual onset time, rise time, peak time, and peak amplitude are studied as functions of T* and τ _m. For very small τ _m , this model is shown to be very similar to previously proposed attenuation models. As τ _m grows past a critical value which depends on T* , the character of the attenuated pulse changes. Seismological implications of this model may be inferred by comparing body wave travel times with a'one second'earth model derived from long-period observations and corrected for attenuation effects assuming a frequency independent Q over the seismic band. From such a comparison we speculate that there may be a gap in the relaxation spectrum of the Earth's mantle for relaxation times shorter than about one second. However, observational constraints from the attenuation of body waves suggest that such a gap might in fact occur at higher frequencies. Such a hypothesis would imply a frequency dependence of Q in the Earth's mantle for short-period body waves.     

Ray asymptotic propagation of the spectra of elastic surface waves in a waveguide with laterally smooth inhomogeneities

Summary. A technique based on ray asymptotics has been developed to propagate complex spectra of elastic normal mode surface waves in a waveguide with material and geometrical properties varying smoothly in the lateral directions. In the technique, the original problem defined in the unstretched coordinates has been transformed into an eiconal equation as well as into a certain number of transport equations defined in stretched coordinates.
The solution of the eiconal equation is equal to the solution of the eigenproblem of the eiconal operator A₀. Due to the self-adjointness of A₀, in each of the relevant local inner product spaces, LIPS, the solution of the eigenproblem, A₀ψ= v ψ results in the set { v _t} of real local eigenvalues and in the orthonormal system {ψ_t} of local eigenvectors.
As the Hamiltonian function of an initial value problem, each eigenvalues gives birth to a bicharacteristic curve as well as to the related ray. The introduction of the rays induces connections between the vertical cross-sections of the waveguide.
Finally, for each asymptotic order j , the LIPS-valued transport equations are reduced to a set of matricial propagation equations in the local spectral amplitude vectors, LSAVs. Consequently, a knowledge of the initial conditions at a vertical cross-section makes it possible to propagate the LSAVs along the rays of the relevant modes. However, to complete the propagation one needs, in addition to the initial values, information about certain additional quantities, non-diagonal terms of order j , diagonal terms of orders lower than j and the auxiliary boundary terms of orders from 1 to j . The treatment has been completed by the propagation of the modal phases along the relevant rays.     

The Backus–Gilbert approach to the 3-D structure in the upper mantle – II. SH and SV velocity

Summary. Phase velocity variations obtained in the previous paper are inverted by the Backus–Gilbert method for the velocity structure of the upper mantle. Spheroidal modes and toroidal modes in the period range of 125–260 s are used in the inversion. The data cannot constrain all six parameters in a transversely isotropic medium and we chose to perturb only two parameters, SH and SV velocities. SV velocities are resolved between the depths of about 200 and 400 km and SH velocities between 0 and 200 km. Resolution kernels have half-peak widths of about 200–300 km in depth, becoming broader for deeper target depths. SV velocity kernels show secondary peaks near the surface of the Earth, with widths varying from 50 to 100 km. The deeper the target depths, the wider the secondary peaks near the surface. SH velocity kernels do not possess such secondary peaks. The trade-off between SV and SH velocities is small. SV velocity is essentially determined by spheroidal modes and SH velocity by toroidal modes. Because of the broad width of the resolution kernels, the structure in the resolved region is difficult to detect from our data set; for example the differences in SV velocity structure between 250 and 350 km or the differences in SH velocity between 100 and 200 km are difficult to distinguish. Considering the horizontal resolution of about 2000 km, obtained in the previous paper, averaging kernels for 3-D structure are quite elongated in the horizontal dimension.     

Crustal structure across the Grand Banks–Newfoundland Basin Continental Margin – II. Results from a seismic reflection profile

New multichannel seismic reflection data were collected over a 565 km transect covering the non-volcanic rifted margin of the central eastern Grand Banks and the Newfoundland Basin in the northwestern Atlantic. Three major crustal zones are interpreted from west to east over the seaward 350 km of the profile: (1) continental crust; (2) transitional basement and (3) oceanic crust. Continental crust thins over a wide zone (∼160 km) by forming a large rift basin (Carson Basin) and seaward fault block, together with a series of smaller fault blocks eastwards beneath the Salar and Newfoundland basins. Analysis of selected previous reflection profiles (Lithoprobe 85-4, 85-2 and Conrad NB-1) indicates that prominent landward-dipping reflections observed under the continental slope are a regional phenomenon. They define the landward edge of a deep serpentinized mantle layer, which underlies both extended continental crust and transitional basement. The 80-km-wide transitional basement is defined landwards by a basement high that may consist of serpentinized peridotite and seawards by a pair of basement highs of unknown crustal origin. Flat and unreflective transitional basement most likely is exhumed, serpentinized mantle, although our results do not exclude the possibility of anomalously thinned oceanic crust. A Moho reflection below interpreted oceanic crust is first observed landwards of magnetic anomaly M4, 230 km from the shelf break. Extrapolation of ages from chron M0 to the edge of interpreted oceanic crust suggests that the onset of seafloor spreading was ∼138 Ma (Valanginian) in the south (southern Newfoundland Basin) to ∼125 Ma (Barremian–Aptian boundary) in the north (Flemish Cap), comparable to those proposed for the conjugate margins.