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.     

Rayleigh-wave dispersion function for a transversely isotropic layered spherical earth using the Thomson-Haskell matrix method

We consider the two coupled differential equations of the two radial functions appearing in the displacement components of spheroidal oscillations for a transversely isotropic (TI) medium in spherical coordinates. Elements of the layer matrix have been explicitly written—perhaps for the first time—to extend the use of the Thomson-Haskell matrix method to the derivation of the dispersion function of Rayleigh waves in a transversely isotropic spherical layered earth. Furthermore, an earth-flattening transformation (EFT) is found and effectively used for spheroidal oscillations. The exponential function solutions obtained for each layer give the dispersion function for TI spherical media the same form as that on a flat earth. This has been achieved by assuming that the five elastic parameters involved vary as r ^p and that the density varies as r ^p-2, where p is an arbitrary constant and r is the radial distance. A numerical illustration with p = - 2 shows that, in spite of the inhomogeneity assumed within layers, the results for spherical harmonic degree n , versus time period T , obtained here for the Primary Reference Earth Model (PREM), agree well with those obtained earlier by other authors using numerical integration or variational methods. The results for isotropic media derived here are also in agreement with previous results. The effect of transverse isotropy on phase velocity for the first two modes of Rayleigh waves in the period range 20 to 240 s is calculated and discussed for continental and oceanic models.     

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.     

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.     

Point sources of elastic deformation: elementary sources, dynamic displacements

We construct a catalogue of all the possible elementary point sources of seismic waves. There are three general classes of sources, two spheroidal and one toroidal. We consider excitation functions for these point-like sources as well as for sources of finite size in far-, intermediate- and near-field for an infinite homogeneous isotropic medium. The sources corresponding to seismic-moment tensors for the second-, third- and fourth-ranks are considered in more detail; we identify 10 different seismic sources in this range: one monopole, two or three dipoles, three quadrupoles, etc. For the step-function of the scalar seismic-moment release, the amplitude spectrum for the third-rank sources is proportional to the angular frequency ω in the region below the corner frequency ω _cr. The fourth-rank sources have an ω ² spectrum in the same range. The possibility of separate and simultaneous inversion of seismic body-wave data and static deformation data for sources of different order is discussed. Some equivalent-force moment higher-rank sources are 'shielded' by lower-rank sources of the same order; the former sources cannot be inverted from seismic data without additional assumptions. Because of their simple radiation pattern, the lower order multipoles, i.e. the monopole and dipoles, are the first sources other than the double-couple which should be considered for inversion.     

Long-period corrections to body waves: theory

Summary. The usual asymptotic methods used to correct the high-frequency solutions of the wave equation are unsatisfactory as they do not give the low-frequency, partial reflections expected from a region of high velocity gradient. A new iterative solution is obtained which uses the first term of the Langer asymptotic expansion as the zeroth iterate. This satisfactorily gives the partial reflections from a region of high velocity gradient, even when they are generated near the turning point of the ray. Although the results are somewhat complicated in the frequency domain, in the time domain all types of wave interaction are described by six universal time functions. For any problem, these functions are scaled in time according to the depth of the interaction, and in strength according to the magnitude of the coupling parameter. Numerical results and approximations are given for these functions. Coupling parameters are investigated for acoustic and elastic waves in a plane model, and acoustic and elastic-gravitational waves in a spherical model. The same universal time functions allow the excitation of elastic waves to be studied when the source is in a region of high velocity gradient or is near the wave's turning point. Results are given for a moment tensor, point source in plane and spherical models.     

Ray-theoretical relations between intrinsic and apparent dissipation factors in the Earth

Summary. A ray-theoretical approach is successfully employed for obtaining the relations between the intrinsic dissipation factors of P - and S -waves, Q _α and Q _β, and the apparent dissipation factors of toroidal and spheroidal oscillations, Q _Tor and Q _SphThese are given in quite simple forms and are expressed, to zero-order approximatiom which neglects the effect of reflection of waves at an interface and/or at a free surface), as where α and β are the P - and S -wave velocities and the integrals are evaluated long the surface to surface P - and S -rays.
Some numerical comparisons support the validity of these formulae for higher modes of toroidal and radial oscillations of a realistic earth model.
A first-order approximation reveals that the existence of a discontinuity in an earth induces a systematic fluctuation in normal-mode Q values for any given phase velocity as a function of frequency (or radial mode number).     

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.     

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.     

Wave-induced responses in a fluid-filled poro-elastic solid with a free surface—a boundary layer theory

Summary. Wave-induced stress in a porous elastic medium is studied on the basis of Biot's linearized theory which is a special case of the mixture theory. For sufficiently high frequencies which are pertinent to ocean waves and seismic waves, a boundary layer of Stokes' type is shown to exist near the free surface of the solid. Outside the boundary layer, fluid and the solid skeleton move together according to the laws of classical elasticity for a single phase. This division simplifies the analysis of the equations governing the two phases; and several examples of potential interest to geophysics and foundation mechanics are treated analytically.     

High-frequency radiation from crack (stress drop) models of earthquake faulting

We present a theory for the radiation of high-frequency waves by earthquake faults. We model the fault as a planar region in which the stress drops to the kinematic friction during slip. This model is entirely equivalent to a shear crack. For two-dimensional fault models we show that the high frequencies originate from the stress and slip velocity concentrations in the vicinity of the fault's edges. These stress concentrations radiate when the crack expands with accelerated motion. The most efficient generation of high-frequency waves occurs when the rupture velocity changes abruptly. In this case, the displacement spectrum has an ω^-2 behaviour at high frequencies. The excitation is proportional to the intensity of the stress concentration near the crack tips and to the change in the focusing factor due to rupture velocity. We extend these two-dimensional results to more general three-dimensional fault models in the case when the rupture velocity changes simultaneously on the rupture front. Results are similar to those described for two-dimensional faults. We apply the theory to the case of a circular fault that grows at constant velocity and stops suddenly. The present theory is in excellent agreement with a numerical solution of the same problem.
Our results provide upper bounds to the high-frequency radiation from more realistic models in which rupture velocity does not change suddenly. The ω^-2 is the minimum possible decay at high frequencies for any crack model of the source.     

Effect of 3-D viscoelastic structure on post-seismic relaxation from the 2004 M= 9.2 Sumatra earthquake

The 2004 M = 9.2 Sumatra–Andaman earthquake profoundly altered the state of stress in a large volume surrounding the ∼1400 km long rupture. Induced mantle flow fields and coupled surface deformation are sensitive to the 3-D rheology structure. To predict the post-seismic motions from this earthquake, relaxation of a 3-D spherical viscoelastic earth model is simulated using the theory of coupled normal modes. The quasi-static deformation basis set and solution on the 3-D model is constructed using: a spherically stratified viscoelastic earth model with a linear stress–strain relation; an aspherical perturbation in viscoelastic structure; a 'static' mode basis set consisting of Earth's spheroidal and toroidal free oscillations; a "viscoelastic" mode basis set; and interaction kernels that describe the coupling among viscoelastic and static modes. Application to the 2004 Sumatra–Andaman earthquake illustrates the profound modification of the post-seismic flow field at depth by a slab structure and similarly large effects on the near-field post-seismic deformation field at Earth's surface. Comparison with post-seismic GPS observations illustrates the extent to which viscoelastic relaxation contributes to the regional post-seismic deformation.     

Range of validity of seismic ray and beam methods in general inhomogeneous media – I. General theory

Summary. The limitations of asymptotic wave theory and its geometrical manifestations are newly formalized and scrutinized. Necessary and sufficient conditions for the existence of acoustic and seismic rays and beams in general inhomogeneous media are expressed in terms of new physical parameters: the threshold frequency ω₀ associated with the P/S decoupling condition, the cut-off frequency ω_c associated with the radiation-zone condition, the total curvature of the wavefront and the Fresnel-zone radius.
The analysis is facilitated with the introduction of a new ancillary functional – the hypereikonal which is capable of representing ordinary as well as evanescent waves. The hypereikonal is the natural extension of the eikonal theory.
With the aid of the above new parameters, simple conditions are obtained for the decoupled far field, the decoupled near field, two point dynamic ray tracing, paraxial wavefields and Gaussian beams.     

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.     

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.     

Generation of complete theoretical seismograms for SH—

Summary. We report the initial results of our attempts to obtain theoretical seismograms for direct comparison with the experimental time series obtained with the long-period instruments of the WWSSN. The entire theoretical seismogram, including both body waves and surface waves, can be generated for a spherical, anelastic earth by simple inverse Fourier transformation of the sum of the propagating fundamental and higher-mode surface waves. The key to success in reproducing the WWSSN records involves the number of modes, and the minimum period used in these computations; here we use eight modes and a minimum period of 2 s. Efficient computational algorithms make it possible to handle up to 2000 frequency points for each mode; approximately 200 layers are used to model the radial heterogeneity of the earth; attenuation is treated exactly. Examples are given of the SH theoretical seismograms resulting from dislocation sources buried at various depths in the Earth.     

Surface vertical displacements, potential perturbations andgravity changes of a viscoelastic earth model induced by internal point dislocations

Using the viscoelastic correspondence principle, we utilize the surface coseismic spheroidal deformation fields (i.e. vertical displacements, potential perturbations and gravity changes) of SNREI earth models caused by four typical types of point dislocation, derived by Sun & Okubo (1993 ), to deduce the fundamental formulas for spheroidal fields relevant to viscoelastic earth models. In computations, we employ a strike-slip dislocation on a vertical plane buried at the bottom of the lithosphere to estimate the maximal viscous relaxation responses to this kind of source that possibly exist on the surface of the earth. We take the seismic moment as 10²²  N  m, which is characteristic of an average large earthquake. The numerical results demonstrate that, if we take the viscosity as 10¹⁹  Pa  s in the asthenosphere, and 10²¹  Pa  s in the other mantle layers, the rates of surface vertical displacements and gravity changes within about 2.5° for the 10 postseismic years are respectively 1.5–8.1  cm  yr⁻¹ and 4.0–14.9  μgal  yr⁻¹ : the viscous relaxation for this mantle viscosity profile proceeds much faster than for a constant mantle viscosity of 10²¹  Pa  s.     

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.