On high-frequency spheroidal modes and the structure of the upper mantle

Summary. The asymptotic properties of spheroidal mode dispersion at high frequency for fixed phase velocity are related to the intercept times τ_β( p ) for P and S waves. If the mode eigenfrequency and the ratio of horizontal to vertical displacement at the surface for the mode are known τ_α( p ) and τ_β( p ) may be separately estimated. If discontinuities exist in the velocity model then 'solotone' effects occur, in frequency at fixed slowness, and in τ_α( p ), τ_β( p ) estimated from the mode dispersion as a function of slowness. The coupling of P and S waves in the spheroidal modes means that the interaction of P waves with upper-mantle discontinuities affects also the estimates of the S wave τ_β( p ) values for which the corresponding turning points lie in the lower mantle. The asymptotic formalism also shows that sharp pulses formed by superposition of spheroidal modes correspond to multiple PS reflections.
A study of τ_α( p ), τ_β( p ) estimates derived from spheroidal modes with periods from 45–50s, calculated for model 1066B, shows that even in the presence of strong upper-mantle discontinuities the errors in intercept time are only about one-tenth of a period. The asymptotic properties may there-for provide a useful means of estimating intercept times from modes with a few seconds period as a supplement to travel-time methods.     

Receiver functions at the stations of the German Regional Seismic Network (GRSN)

Several years of broad-band teleseismic data from the GRSN stations have been analysed for crustal structure using P -to- S converted waves at the crustal discontinuities. An inversion technique was developed which applies the Thomson-Haskell formalism for plane waves without slowness integration. The main phases observed are Moho conversions, their multiples in the crust, and conversions at the base of the sediments. The crustal thickness derived from these data is in good agreement with results from other studies. For the Gräfenberg stations, we have made a more detailed comparison of our model with a previously published model obtained from refraction seismic experiments. The refraction seismic model contains boundaries with strong velocity contrasts and a significant low-velocity zone, resulting in teleseismic waveforms that are too complicated as compared to the observed simple waveforms. The comparison suggests that a significant low-velocity zone is not required and that internal crustal boundaries are rather smooth.     

Slowness-backazimuth weighted migration: A new array approach to a high-resolution image

We have developed a new array method combining conventional migration with a slowness-backazimuth deviation weighting scheme. All seismic traces are shifted based on the theoretical traveltime of the scattered wave from specific gridpoints in a 3-D volume. Observed slowness and backazimuth are calculated for each raypath and compared with theoretical values in order to estimate slowness and backazimuth deviations. Subsequently, stacked energy calculated by a conventional migration method is weighted by the slowness and backazimuth deviations to suppress any arrival energy whose slowness and backazimuth are inconsistent with the expected theoretical values. This new method was applied to two P- wave data sets which comprise (1) underside reflections at the 410 and 660 km mantle discontinuities and (2) D" reflections as well as their corresponding synthetic data sets. The results show that the weighting scheme dramatically increases the resolution of the migrated images and enables us to obtain well-constrained, focused images, making upper-mantle discontinuities and D" reflections more distinct by reducing their surrounding energy.     

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.     

Moho-offset beneath northern Scotland

Summary. Independent sets of LISPB data are presented, having as a common feature a local lateral variation in the Moho beneath northern Britain.
The evidence for this step-like feature on the Moho is taken from P and S travel times and from PS reflection times and phase velocities.
Another important observation is that a high-frequency Moho refraction is generated north of the step, whilst the refraction from the step region and south of it contains only low frequencies. This leads to the assumption that a change in the Moho structure exists in the same region as where the step has been deduced from travel-time interpretation. A numerical model is presented which generates a high-frequency refracted wave from an input signal which contains low- and high-frequency spectral energy.
The significance of this offset on the Moho, in particular its relationship to surface tectonics, is critically discussed.     

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.     

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.     

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

Summary. In examining the effect of discontinuities in the Earth's interior on free oscillations, McNabb, Anderssen & Lapwood derived an equation for the asymptotic behaviour of torsional overtone eigenfrequencies of a discontinuous earth model, the constants in their equation being explicitly determined only for the case of one internal discontinuity. Since Brune's phase correlation method for the evaluation of eigenfrequencies from body-wave data implies a ray-mode duality only for continuous earth models, it is desirable to justify the McNabb et al. formulation from the point of view of ray theory.
By a novel method of ray analysis, Wang, Cleary & Anderssen showed that, for earth models with a single discontinuity between the Earth's surface and the core—mantle boundary, the McNabb et al. formulation can be derived from an adaptation of Brune's method to multiply reflected SH body waves recorded at small epicentral distances. In this paper, the technique of Wang et al. is extended to derive the McNabb et al. formulation (with constants explicitly determined) for the general case of earth models with N discontinuities. This establishes a basis for a ray-mode duality for discontinuous earth models.     

Oscillation, Nutation and Wobble of an Elliptical Rotating Earth with Liquid Outer Core

We have constructed a general first-order theory describing those small oscillations of a rotating elliptical earth that are affected by the presence of a liquid outer core. The theory is applicable to free core oscillations and earth tides. Care has been taken to include the effects of the wobble or nutation due to the rotation of the outer core relative to the solid earth. On the basis of the theory the free spheroidal modes of degree 2 and order 1 have been investigated. We have searched for and listed undertones with periods less than 28 hr. No upper limit to the eigenperiods has been detected. It is shown that stable, unstable and neutral polytropic cores are capable of free oscillation. At a period close to the sidereal day the spheroidal mode is accompanied by rigid rotation of the liquid outer core with respect to the solid earth. This is the well-known diurnal wobble of the Earth. It appears probable that the diurnal wobble is one of a class of similar wobbles that involve large toroidal motions in the outer core. Finally, the amplitudes of the 18·6-yr principal nutations has been computed. Excellent agreement is found with observed values.     

Variationsl solution of long-period oscillations of the Earth

Summary The linearized equation of motion for the slightly elliptical rotating earth is obtained and using Phinney & Burridge's generalized spherical harmonics, the variational principle is derived for the normal mode oscillations of the Earth. The numerical solutions of two earth models 1066B and B1S6 are searched by minimizing the energy functional for the terrestrial spectral range longer than the lowest order free oscillation. The periods of core modes computed for the earth model B1S6, with stably stratified outer core, ranges from about 4 to 13hr and the periods for the 1066B are much more spread without clustering around the periods of 6 and 12 hr as in B1S6. The results for the earth model 1066B indicate that an outer core can support long-period oscillations even when it is not stably stratified. The Chandler wobble periods obtained are 402.3 day for B1S6 and 402.7 day for 1066B.     

Determination of Moho dip using PS reflections

Summary. A clear PS reflection was recorded in the northern section of the LISPB line (shot-point N1, profile ALPHA). Travel times of PS and PP phases reflected at the same part of the Moho were used to model the lower crustal structure. The observed high apparent velocity of the PS phase (8.4 km/s) is interpreted as due to a localized Moho dip of 7 per cent (4°) towards the north. This is supported by observations of Pn refractions from other shots. It is shown that PS reflections could be very useful for determination of Moho dips.     

An asymptotic approach to the inversion of free oscillation data

Summary. A technique is suggested for the non-linearized inversion of subsets of free oscillation periods which are short enough that their asymptotic properties may be exploited. The general background of the method for monotonic and non-monotonic velocity distributions are described and applications to torsional and spheroidal free oscillation data are suggested. The advantages of joint analysis of travel times and free oscillation periods as a single data stream for non-linearized inversion are discussed.     

A physical explanation of the static core paradox

Summary. The introduction of Rayleigh friction and Newtonian cooling into the dynamical problem of determining the excitation of the normal modes of oscillation of an earth model with a fluid core by a transient earthquake source is shown to provide a fully satisfactory resolution and a clear physical explanation of the difficulties and paradoxes which have arisen in previous treatments of the corresponding static deformation problem. The source of the previous difficulties is that the dissipation-free limit is associated with an essential singularity in the static response, unless the stratification in the core is neutral. This singularity, in turn, exists because the eigenfrequency spectrum of any earth model with a non-neutrally stratified core has an accumulation point at zero frequency.     

Seismic ray theory for lithospheric structures with slight lateral variations

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In this paper, the method of small perturbations is applied to the ray and energy transport equations in an investigation of the effect of weak inhomogeneities on the propagation of seismic rays through a layer of fixed thickness. Just as Aki et al . have used travel-time residuals to infer first-order velocity perturbations in their block-modelling procedure, it is proposed that surface slowness and amplitude data may be used to give additional information about the structure of the velocity perturbations beneath the observer.     

Source mechanism of the deep Colombian earthquake of 1970 July 31 from the free oscillation data

Summary. The method proposed by Mendiguren to determine the source parameters from free oscillation data is applied to the 1970 July 31 deep Colombian earthquake. The results indicate a source propagating horizontally for about 150 km along the lithosphere and cutting across its width. The slab behaves as a guide for source propagation. The horizontal propagation velocity is determined as 3.8 km/s. The intensity of the source grew proportionately to the second power of the propagation distance. This rate of source intensity growth may be interpreted either by a fan-shaped fault model or by a cone-shaped volume source. The average slip and stress drop are estimated as 360 cm and 300 bar for the fault model. For the volume source model the transformational shear strain and stress are estimated as 11 × 10⁻⁵ and 160 bar. There is no evidence of a double couple radiation preceding the P origin time. It is shown that the isotropic and deviatoric components of the moment tensor cannot be uniquely resolved when only observations of a single mode are available. It is observed that, from a statistical basis, the available ₀ S_n data for Colombian shock can be equally well explained by a pure deviatoric source model or by a source model including an isotropic component. Numerical experiments indicate that the inclusion of higher mode data does not change this situation. But, on the other hand, numerical experiments show that the available data and the scheme used for the inversion would not result in a solution including an artificial implosive component if the actual source were pure deviatoric. If the departure from a pure deviatoric source is produced by noise, it has to be non-random, as it could be produced by lateral heterogeneities not included in the inversion scheme.     

The influence of a density stratification and of an incompressibility modulus on the spectrum of core modes

The concept of a deformation of a simple, non-rotating, spherically symmetric earth model with a fluid outer core, although it is a highly artificial physical situation, provides a useful computational algorithm that allows one lo determine analytically modes of vibration without any Love-number theory. In particular, on these analytically determined modes, we impose regularity conditions at the centre and boundary conditions at the surface, as well as conditions of continuity at the inner-core-outer-core boundary and at the core-mantle boundary. They lead to an eigenvalue equation for the frequency of oscillation. The range of frequencies obtained in this way for different earth models gives an indication of the influence of compressibility and non-homogeneity on the spectrum of eigenfrequencies.     

Inverse problems for torsional modes

Summary. We consider a spherically symmétric, non-rotating earth consisting of an isotropic, perfect elastic material where the density and the S -wave velocity may have one or two discontinuities in the upper mantle. We show that given the velocity throughout the mantle and the crust and given the density in the lower mantle, then the frequencies of the torsional oscillations of one angular order (one torsional spectrum), determine the density in the upper mantle and in the crust uniquely. If the velocity is known only in the lower mantle, then the frequencies of the torsional oscillations of two angular orders uniquely determine both the density and the velocity in the upper mantle and in the crust. In particular, the position and size of the discontinuities in the density and velocity are uniquely determined by two torsional spectra.     

The effect of a second-order velocity discontinuity on elastic waves near their turning point

Summary. The rather abrupt changes in velocity gradient which have sometimes been proposed, notably in the upper mantle and near the base of the mantle, have an effect equivalent to that of one or more second-order discontinuities, where partial reflection occurs due to a change in curvature of the wavefront across these discontinuities. The effect is ignored in the classical WKBJ approximation to the wave functions, but it can be explicitly demonstrated by applying the extended WKBJ method (Langer's approximation) to a piecewise smooth layered model. For the purpose of this study it is convenient to represent the response of such a modelby a generalized reflection coefficient. For a model of one or a system of several second-order discontinuities (approximating a change in velocity gradient over a finite depth interval), the reflection coefficient can be perhaps surprisingly large for long-period waves near their turning point. It is shown that this effect can significantly alter the amplitude decay of SH waves diffracted around the core, in models where a change in velocity gradient near the core—mantle boundary constitutes a low-velocity zone at the base of the mantle; such models have recently been proposed. With the same velocity gradients, the effect on P diffraction is less important. The results for SH diffraction in these models support the conclusion that a small amplitude decay must be explained by a velocity decrease with depth, i.e. a low-velocity zone at the base of the mantle.     

Generalized Born scattering of elastic waves in 3-D media

It is well known that when a seismic wave propagates through an elastic medium with gradients in the parameters which describe it (e.g. slowness and density), energy is scattered from the incident wave generating low-frequency partial reflections. Many approximate solutions to the wave equation, e.g. geometrical ray theory (GRT), Maslov theory and Gaussian beams, do not model these signals. The problem of describing partial reflections in 1-D media has been extensively studied in the seismic literature and considerable progress has been made using iterative techniques based on WKBJ, Airy or Langer type ansätze. In this paper we derive a first-order scattering formalism to describe partial reflections in 3-D media. The correction term describing the scattered energy is developed as a volume integral over terms dependent upon the first spatial derivatives (gradients) of the parameters describing the medium and the solution. The relationship we derive could, in principle, be used as the basis for an iterative scheme but the computational expense, particularly for elastic media, will usually prohibit this approach. The result we obtain is closely related to the usual Born approximation, but differs in that the scattering term is not derived from a perturbation to a background model, but rather from the error in an approximate Green's function. We examine analytically the relationship between the results produced by the new formalism and the usual Born approximation for a medium which has no long-wavelength heterogeneities. We show that in such a case the two methods agree approximately as expected, but that in a media with heterogeneities of all wavelengths the new gradient scattering formalism is superior. We establish analytically the connection between the formalism developed here and the iterative approach based on the WKBJ solution which has been used previously in 1-D media. Numerical examples are shown to illustrate the examples discussed.     

Effects of dissipation and a liquid core on forced nutations in Hamiltonian theory

In a previous paper, the authors considered the free rotation of an earth model composed of a rigid mantle and a liquid core in the presence of dissipation and under the Hamiltonian formalism, obtaining analytical expressions for the free nutation modes.
In this paper we treat the forced motion. Approximate analytical solutions are worked out by means of Hori's perturbation method, the free solutions obtained in the former paper playing the role of the unperturbed solutions required in the application of the method. These solutions are consistent in the sense that, with the usual terminology, the rigid body solutions and the complex transfer functions are calculated with the same parameters.
Besides in-phase terms, the dissipation at the core–mantle boundary studied in this paper gives rise to out-of-phase terms. From a qualitative perspective, we discuss the issue of the resonance in this context. The presence of dissipation changes dramatically the character of the FCN wobble; that is, it is no longer a regular oscillation but a damped one. A strict resonance phenomenon cannot take place thereby, since the forcing perturbations are oscillations with a real (non-complex) frequency.