Ray-mode duality for SH waves in earth models with crust and mantle discontinuities—I. The case of one discontinuity

Summary. The method of phase correlation devised by Brune for the evaluation of overtone eigenfrequencies of the Earth from body-wave data involves the explicit use of a ray-mode duality for strictly continuous earth models. Consequently, investigation of the application of the ray-mode duality approach to models containing discontinuities is desirable.
McNabb, Anderssen & Lapwood have obtained an equation for the behaviour of overtone eigenfrequencies from discontinuous earth models. By means of a novel method for the decomposition of multiply-reflected body waves, it can be shown that, for earth models with a single discontinuity between the surface and the core—mantle boundary, the McNabb et al. formulation can be derived from an adaptation of Brune's formula to multiply-reflected SH body waves at small epicentral distances. This establishes a basis for a ray-mode duality for models with a single discontinuity.     

Asymptotic overtone structure in eigenfrequencies of torsional normal modes of the Earth: a model study

Summary. The study of the asymptotic behaviour of eigenfrequencies of torsional modes of the Earth is of interest in the problem of the existence of discontinuities in the Earth's interior and the determination of their depth and scale, since the solotone effect, which is a persistent oscillatory cohponent in the asymptotic overtone structure, is very sensitive to the presence of discontinuities. The asymptotic behaviour of torsional eigenfrequencies of the second order differential equation for the Earth's free oscillations can be compared with that obtained from eigenfrequencies evaluated from synthetic SH-wave seismograms by Brune's phase correlation method, using various earth models. The solotone effect that appears in the former for discontinuous models can be explained in terms of multiple reflections fram the discontinuities, and can be reconstructed from synthetic SH -wave pulses arising from these reflections. Its properties vary systematically with the depth and the scale of discontinuities and can be superposed for several discontinuities.     

Asymptotic eigenfrequencies of the Earth's normal modes

We derive asymptotic formulae for the toroidal and spheroidal eigenfrequencies of a SNREI earth model with two discontinuities, by considering the constructive interference of propagating SH and P-SV body waves. For a model with a smooth solid inner core, fluid outer core and mantle, there are four SH and 10 P-SV ray parameters regimes, each of which must be examined separately. The asymptotic eigenfrequency equations in each of these regimes depend only on the intercept times of the propagating wave types and the reflection and transmission coefficients of the waves at the free surface and the two discontinuities. If the classical geometrical plane-wave reflection and transmission coefficients are used, the final eigenfrequency equations are all real. In general, the asymptotic eigenfrequencies agree extremely well with the exact numerical eigenfrequencies; to illustrate this, we present comparisons for a crustless version of earth model 1066A.     

Retrieving shear velocity structure from high-frequency toroidal modes of the Earth

Summary. For a smooth earth model, observations of a set of high-frequency toroidal modes at fixed slowness yield only a single piece of information, the tau value for that slowness. In this note, a procedure for obtaining the shear velocity structure from free oscillation data for an earth model with velocity discontinuities is developed, based on the method of tau inversion. The information content of the high-frequency modes is greater in this case, and the nature and depths of the discontinuities may be deduced. It is shown, for the real Earth, that the tau values obtained from free oscillation data are affected significantly by the presence of the Moho, but a simple iterative scheme may be used to remove this contamination. Brune's method of deducing mode frequencies from body wave pulses is shown to produce significant errors for a model with a pronounced Moho discontinuity, and the same iterative scheme may also be employed to correct for this effect.     

Material versus isobaric internal boundaries in the Earth and their influence on postglacial rebound

Most previous earth models used to calculate viscoelastic relaxation after the removal of the Late Pleistocene ice loads implicitly assume that there is no exchange of mass across the mantle density discontinuities on periods of tens of thousands of years (the material boundary formulation). In the present study, simple incompressible models are used to determine the Earth's behaviour in the case where the density discontinuity remains at a constant pressure rather than deforming with the material (the isobaric boundary formulation). The calculation of the movement of the boundary is more rigorous than in earlier studies and uses the local incremental pressure calculated at the depth of the boundary and allows for the vertical deformation caused by the change in volume as material changes phase. It is shown that the buoyancy modes associated with the density discontinuities decrease in strength and increase in relaxation time analogous to what results when the density contrast is reduced. Also, two viscoelastic modes arise from an isobaric boundary, which is also predicted when there is a contrast in rigidity or viscosity across a material boundary. The difference in predicted radial deformation between the isobaric boundary model and the material boundary model is largest for long-wavelength loads for which the material incremental pressure at depth is largest. If the isobaric boundary model is appropriate for the treatment of the mineral phase changes in the mantle on glacial rebound timescales, then previous inferences of the deep-mantle to shallow-mantle viscosity ratio based on large-scale deformation (spherical harmonic degree < 10) of the Earth and including data from the early part of the glacio-isostatic uplift are too small.     

A search for Chandler and nearly diurnal free wobbles using Liouville equations

Summary. The basic equations describing the dynamical effects of the Earth's fluid core (Liouville, Navier-Stokes and elasticity equations) are derived for an ellipsoidal earth model without axial symmetry but with an homogeneous and deformable fluid core and elastic mantle.
We develop the balance of moment of momentum up to the second order and use Love numbers to describe the inertia tensor's variations. The inertial torque takes into account the ellipticity and the volume change of the liquid core. On the core—mantle boundary we locate dissipative, magnetic and viscous torques. In this way we obtain quite a complete formulation for the Liouville equations.
These equations are restricted in order to obtain the usual Chandler and nearly diurnal eigenfrequencies.
Then we propose a method for calculating the perturbations of these eigenfrequencies when considering additional terms in the Liouville equations.     

Modes to replace transitional asymptotic ray fields in a vertically inhomogeneous earth model

Summary. Asymptotic ray theory (ART) fails in transition regions near critically reflected, bottom glancing or caustic-forming rays in a vertically inhomogeneous layered earth. These deficiencies are repaired here by replacing the transitional ray fields with guided modes plus truncation remainders. Exact ray-mode equivalences and their high-frequency asymptotic approximations are formulated, and their validity and efficiency are verified by numerical comparisons for SH motion in a two-layer earth model comprised of an inhomogeneous sediment above an homogeneous semi-infinite bedrock.     

A Theory of Toroidal Core Oscillations of the Earth

A rotating incompressible fluid bounded by two concentric spherical rigid surfaces can exhibit purely toroidal free oscillations. The eigenfrequencies are fractions of the angular frequency of rotation. If the bounding surfaces are slightly ellipsoidal, secondary spheroidal fields become existent, and in general, a free mode splits into a doublet with one of which exists only when the inner bounding surface is present.
For the real earth, the compressibility of the outer core, the elasticity of the solid earth, and the self-gravitation of the entire earth modify the toroidal core oscillations. The present treatment gives explicitly the effects of these parameters on the eigenfrequencies.     

The effect of discontinuities in density and shear velocity on the asymptotic overtone structure of torsional eigenfrequencies of the Earth

Summary. This paper pursues theoretically and comments on results obtained by Wang, Gettrust & Cleary from a model study of the asymptotic overtone structure of torsional eigenfrequencies of the Earth. It shows that a number of qualitative results, which they derive about the dependence of the asymptotic overtone structure on the position and nature of discontinuities in density and shear velocity, can be derived quantitatively.     

Matrix Green's functions for array-type sources and receivers in multiwave layered media

Summary. A previous formulation (Lu, Felsen & Kamel) of source-excited wave propagation in a multiwave layer is here extended to multiple layers, each of which may propagate different multiple wave species, and to simultaneous excitation and detection at arbitrarily specified multiple levels. Field variables are arranged so as to reveal 'interesting'layers requiring access (for example, those containing a source and/or receiver) but to hide in collective form all other 'uninteresting'layers. An ordering of wave constituents into array vectors provides not only a physically appealing view of the wave phenomena pertaining to array-type source and receiver arrangements but may also furnish numerical advantages. The variety of alternative representations in Lu et al. can be brought to bear directly on the present formulation which is thereby endowed with substantial versatility, especially that embodied within the hybrid ray-mode format.     

Pseudo-bending method for three-dimensional seismic ray tracing in a spherical earth with discontinuities

We present a 'pseudo-bending' approach to 3-D ray tracing in a spherical earth with discontinuities. This method is based on a three-point perturbation associated with a first-order approximation, while Snell's law in curvilinear coordinates is applied at the discontinuities. We demonstrate the computational accuracy and efficiency of the pseudo-bending method in tracing rays for various velocity models by comparing results with analytical solutions and with results from the bending method. The improvement of efficiency is significant, but is reduced as the number of discontinuities increases. Since the bending approach may be computationally unstable in some situations, even though it is exact, the pseudo-bending approach is preferable for automatic calculation of rays.     

Global mapping of upper mantle reflectors from long-period SS precursors

Long-period precursors to SS resulting from underside reflections off upper mantle discontinuities ( SdS where d is the discontinuity depth) can be used to map the global distribution and depth of these reflectors. We analyse 5,884 long-period seismograms from the Global Digital Seismograph Network (1976-1987, shallow sources, transverse component) in order to identify SdS arrivals. Corrections for velocity dispersion, topography and crustal thickness at the SS bounce point, and lateral variation in mantle velocity are critical for obtaining accurate estimates of discontinuity depths. The 410 and 660 km discontinuities are observed at average depths of 413 and 653 km, and exhibit large-scale coherent patterns of topography with depth variations up to 40 km. These patterns are roughly correlated with recent tomographic models, with fast anomalies in the transition zone associated with highs in the 410 km discontinuity and lows in the 660 km discontinuity, a result consistent with laboratory measurements of Clapeyron slopes for the appropriate phase changes. The best resolved feature in these maps is a trough in the 660 km discontinuity in the northwest Pacific, which appears to be associated with the subduction zones in this region. Amplitude variations in SdS arrivals are not correlated with discontinuity depths and probably result from focusing and defocusing effects along the ray paths. The SdS arrivals suggest the presence of regional reflectors in the upper mantle above 400 km. However, only the strongest of these features are above probable noise levels due to sampling inadequacies.     

First-order asymptotics for the eigenfrequencies of the Earth and application to the retrieval of large-scale lateral variations of structure

Summary . We present a variety of examples, showing systematic fluctuations as a function of angular order of measured eigenfrequencies for given normal modes of the Earth. The data are single station measurements from the GEOSCOPE network. Such fluctuations are attributed to departures from the lowest order asymptotic expression of the geometrical optics approximation. We derive first-order asymptotic expressions for the location parameter for all three components of the Earth's motion, by a method based on the stationary phase approximation and geometric relations on the unit sphere.
We illustrate the sensitivity of the fluctuations to the different parameters involved (source parameters, epicentral distance, laterally heterogeneous earth model) with synthetic examples corresponding to GEOSCOPE observations. Finally, we show the results of first attempts at inversion, which indicate that, when the fluctuations are taken into account, more accurate estimates of the great circle average eigenfrequencies can be obtained, and additional constraints put on the structure in the neighbourhood of the great circle.     

Inversion for laterally heterogeneous upper mantle S-wave velocity structure using iterative waveform inversion

In contrast to previous work, which treats the Earth's lateral heterogeneity as an infinitesimal perturbation to a spherically symmetrical starting model, we conduct iterative linearized waveform inversion for the Earth's laterally heterogeneous structure. We use the Direct Solution Method (DSM) (Geller et al. 1990a) to calculate synthetic seismograms and their partial derivatives for a laterally heterogeneous earth model. We invert surface-wave data from the IDA and GEOSCOPE networks. We expand the lateral heterogeneity of rigidity in spherical harmonics up to angular order number 8 and use three parameters to specify the depth dependence of each harmonic, giving us a total of 240 unknowns. The short-wavelength lateral heterogeneity (s = 4, 6 and 8) in the deeper part of the upper mantle obtained by this study differs significantly from M84A. The relative improvement in the variance reduction as compared with model M84A is about 20 per cent for the IDA data and more than 100 per cent for the GEOSCOPE data.     

Spherical prism magnetic effects by Gauss–Legendre quadrature integration

Regional spherical coordinate observations of the Earth's crustal magnetic field components are becoming increasingly available from shipborne, airborne, and satellite surveys. In assessing the geological significance of these data, theoretical anomalous magnetic fields from geologic models in spherical coordinates need to be evaluated. This study explicitly develops the elegant Gauss–Legendre quadrature formulation for numerically modelling the complete magnetic effects (i.e. potential, vector and tensor gradient fields) of the spherical prism. We also use these results to demonstrate the magnetic effects for the crustal prism and to investigate the crustal magnetic effects at satellite altitudes for a large region of the Middle East centred on Iran.     

Imaging upper-mantle discontinuity topography usingunderside-reflection data

This paper presents a method to invert underside-reflection ( P _d P or S _d S arrivals) data for lateral depth variations of upper-mantle discontinuities, combining traveltime and amplitude data. The method greatly improves the resolution of small-scale undulations obtained by existing imaging methods and does not suffer from the long-wavelength biases that are likely to be present in currently available models. Existing inversion methods account for the large size of the Fresnel zone of underside reflections, but not for its complexity, arising from the mini-max traveltime nature of PP- and SS -related waves. This neglect results in long-wavelength artefacts from small-scale undulations of the discontinuities, obscuring true long-wavelength depth variations. The inversion method presented in this paper uses a complex-valued sensitivity kernel, derived from the representation of underside reflections through a Kirchhoff integral formulation. The sensitivity kernel accounts for the varying sensitivity of the waveforms to discontinuity structure over the Fresnel zone. The method is applied to a large, synthetic data set. The data set consists of P _d P amplitudes and traveltimes. The results show that the new inversion method resolves depth variations on a lateral scale that is smaller than the size of the Fresnel zone of individual underside reflections (but larger than the dominant wavelength), retaining the resolution of large-scale variations. The results presented here suggest that the discontinuity depth variations induced by slab penetration of the 670 discontinuity could be resolved by current broad-band P ₆₇₀ P data sets.     

Shallow layer correction for Spectral Element like methods

Today's numerical methods like the Spectral Element Method (SEM) allow accurate simulation of the whole seismic field in complex 3-D geological media. However, the accuracy of such a method requires physical discontinuities to be matched by mesh interfaces. In many realistic earth models, the design of such a mesh is difficult and quite ineffective in terms of numerical cost. In this paper, we address a limited aspect of this problem: an earth model with a thin shallow layer below the free surface in which the elastic and density properties are different from the rest of the medium and in which rapid vertical variations are allowed. We only consider here smooth lateral variations of the thickness and elastic properties of the shallow layer. In the limit of a shallow layer thickness very small compared to the smallest wavelength of the wavefield, by resorting to a second order matching asymptotic approximation, the thin layer can be replaced by a vertically smooth effective medium without discontinuities together with a specific Dirichlet to Neumann (DtN) surface boundary condition. Such a formulation allows to accurately take into account complex thin shallow structures within the SEM without the classical mesh design and time step constraints. Corrections at receivers and source—when the source is located within the thin shallow layer—have been also derived. Accuracy and efficiency of this formulation are assessed on academic tests. The stability and limitations of this formulation are also discussed.     

On the focusing of seismic body waves at the epicentre's antipode

Summary. This note reports on the remarkable focusing of seismic body waves at or near the antipode (Δ= 180°) of an earthquake's epicentre. The particular seismic velocity structure and sphericity of the Earth cause body-wave phases such as P (diff), PKP, PP, PPP, PcPPKP, SKSSKS, SS , etc. to converge individually at antipodal distances after being diffracted, reflected or refracted at discontinuities. This focusing strongly amplifies each signal up to almost one order of magnitude with respect to the normal phase recorded two or more degrees away. Since the signal/noise ratio is enhanced in the same proportion, seismograms at antipodal distances provide clear and strong arrivals of otherwise weak phases. Antipodal monitoring of seismic waves is suggested as a powerful means of exploring the Earth's interior. The study of these 'seismic images' generated at focal points of seismic rays will yield information on the departures from lateral homogeneity and sphericity of the core, as well as stronger constraints on earth models. To interpret the observations correctly, the data must be compared with theoretically generated seismograms. Since the appropriate ray theory equations (see, e.g. Scholte; Gilbert & Helmberger; Richards) are singular at Δ=180°, a corrective measure is taken which provides a formal expression for the wave amplitude that remains finite at the antipode, and reproduces the usual expressions at other distances.     

Glacial isostatic adjustment in Fennoscandia for a laterally heterogeneous earth

Glaciation and deglaciation in Fennoscandia during the last glacial cycles has significantly perturbed the Earth's equilibrium figure. Changes in the Earth's solid and geoidal surfaces due to external and internal mass redistributions are recorded in sequences of ancient coastlines, now either submerged or uplifted, and are still visible in observations of present‐day motions of the surface and glacially induced anomalies in the Earth's gravitational field. These observations become increasingly sophisticated with the availability of GPS measurements and new satellite gravity missions.
Observational evidence of the mass changes is widely used to constrain the radial viscosity structure of the Earth's mantle. However, lateral changes in earth model properties are usually not taken into account, as most global models of glacial isostatic adjustment assume radial symmetry for the earth model. This simplifying assumption contrasts with seismological evidence of significant lateral variations in the Earth's crust and upper mantle throughout the Fennoscandian region.
We compare predictions of glacial isostatic adjustment based on an ice model over the Fennoscandian region for the last glacial cycle for both radially symmetric and fully 3‐D earth models. Our results clearly reveal the importance of lateral variations in lithospheric thickness and asthenospheric viscosity for glacially induced model predictions. Relative sea‐level predictions can differ by up to 10–20 m, uplift rate predictions by 1–3 mm yr⁻¹ and free‐air gravity anomaly predictions by 2–4 mGal when a realistic 3‐D earth structure as proposed by seismic modelling is taken into account.     

Evidence for a seismic discontinuity near 200km depth under a continental margin

Summary. We present evidence for a seismic discontinuity near 200km depth (the Lehmann Discontinuity) under the passive continental margin of northwest Australia, where continental lithosphere merges into oceanic lithosphere. The velocity contrast across the discontinuity is 0.2–0.3 km s^-1, and is similar to the contrast across discontinuities at similar depths in seismic models for purely continental paths to the east under central Australia. The discontinuity has been shown to be present under continents, oceans and now at continental margins, and is probably a worldwide feature.