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  • 1
    Electronic Resource
    Electronic Resource
    [s.l.] : Macmillan Magazines Ltd.
    Nature 402 (1999), S. 26-27 
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] Is the Earth's inner core rotating slightly faster than the mantle and crust? This is a question that has perplexed geophysicists since it was first raised five years ago. The latest study comes from Laske and Masters (page 66 of this issue), who have taken a new approach to the problem and ...
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  • 2
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 123 (1995), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: A necessary and sufficient condition for the validity of surface-wave ray theory is 〈inlineGraphic alt="inline image" href="urn:x-wiley:0956540X:GJI757:GJI_757_mu1" location="equation/GJI_757_mu1.gif"/〉≪ 1, where 〈inlineGraphic alt="inline image" href="urn:x-wiley:0956540X:GJI757:GJI_757_mu2" location="equation/GJI_757_mu2.gif"/〉 is the root-mean-square degree of the phase-velocity perturbation δc, and l is the degree of the equivalent mode nSl or nTl. This condition, which is independent of the number of orbits, is obtained by consideration of the Fresnel area along the surface-wave ray path between the source and receiver. We assess the accuracy of surface-wave ray theory by comparing the phase, arrival angle and amplitude anomalies obtained using the JWKB approximation with the corresponding quantities measured using ‘ground-truth’ synthetic coupled-mode seismograms on models of varying roughness. The JWKB results agree well with the coupled-mode results for model S12_WM13; on a contrived model with slightly rougher variations, however, the agreement deteriorates as the condition 〈inlineGraphic alt="inline image" href="urn:x-wiley:0956540X:GJI757:GJI_757_mu3" location="equation/GJI_757_mu3.gif"/〉≪ 1 is less well satisfied. The misfit of the ray approximation, which is dependent upon the quantity 〈inlineGraphic alt="inline image" href="urn:x-wiley:0956540X:GJI757:GJI_757_mu4" location="equation/GJI_757_mu4.gif"/〉, can be attributed to diffraction and other finite-frequency effects within the Fresnel area along the ray path; these effects are ignored by the JWKB theory but are fully accounted for by the coupled-mode summation. The wavefront smoothing produced by this Fresnel-area averaging limits the resolution of surface-wave inversion studies based upon the JWKB approximation.
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  • 3
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 126 (1996), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We complete our series of studies on the duality between relatively high-frequency normal modes and seismic body waves by explicitly converting the normal-mode summation expressing the elastic response of a spherically symmetric earth model into a body-wave summation. The procedures in this conversion are discussed in detail for a crust-stripped version of model 1066A with a solid mantle, a fluid outer core and a solid inner core. The resulting body-wave responses, the Green tensors representing the various body waves, are in agreement with the well-documented results obtained using geometrical ray theory.The results derived for the spherically symmetric earth model are then used to obtain the first-order body-wave Green tensors in a laterally heterogeneous earth model based upon the Born approximation. The first-order effect of the coupling between normal-mode multiplets is shown to be equivalent to the single scattering of seismic body waves by the lateral heterogeneity, as expected from numerous previous studies. We present compact expressions for the scattering matrix and the body-wave Frechet kernels. A few numerical examples of the Frechet kernels for monochromatic body waves are also provided.
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  • 4
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 112 (1993), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We present a frequency-domain analysis of travelling and standing surface waves on a smooth, laterally heterogeneous Earth model, using a potential representation that is valid everywhere, including in the neighbourhood of surface wave caustics. the Love and Rayleigh wave displacement fields are written in the form uL=k-1LW (-řX▿1)χL and uR=UřχR+k-1RV▿1χR; the quantities U, V and W are the local radial eigenfunctions, kL and kR are the local Love and Rayleigh wavenumbers, and χL and χR are surface wave potentials that vary rapidly on the surface of the unit sphere. the natural normalization condition for the local radial eigenfunctions is cCI1= 1, where c is the local phase velocity, C is the local group velocity and I1 is the local radial kinetic energy integral; with this normalization, the Love and Rayleigh wave potentials satisfy the spherical Helmholtz equations ▿21χL + k2LχL = 0 and ▿21χR + k2RχR = 0. Standing wave eigenfunctions χL and χR can be determined either by solving a truncated matrix eigenvalue problem or by using the EBK semi-classical method; the results incorporate multiplet coupling along a single dispersion branch but ignore cross-branch coupling. the theory allows for slowly varying topography of the Earth's surface and the core-mantle boundary, and incorporates the effect of self-gravitation.
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  • 5
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 115 (1993), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: 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.
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  • 6
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 109 (1992), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We present a JWKB theory that describes the propagation of well-dispersed Love and Rayleigh wavegroups on a smooth, laterally heterogeneous Earth model. The analysis is based upon an averaged Lagrangian which yields local Love and Rayleigh eigenfunctions, local dispersion relations, and conservation laws for the surface wave energy. The local dispersion relations determine the surface wave trajectories, and the energy equations determine the surface wave amplitudes. The amplitude of a surface wavegroup varies in time as a result of both dispersion and geometrical spreading. The theory allows for smooth topography on the Earth's surface and any internal discontinuities, and incorporates the effect of self-gravitation.
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  • 7
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 109 (1992), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We present a JWKB theory for the propagation of monochromatic Love and Rayleigh waves on a smooth, laterally heterogeneous Earth model. The analysis is based upon a slowly varying Lagrangian which yields local Love and Rayleigh eigenfunctions, local dispersion relations, and transport equations which determine the variation in surface wave amplitude along a ray. The amplitude of a monochromatic Love or Rayleigh wave varies only as a result of geometrical spreading; the amplitude diverges and the phase is shifted by /2 each time the wave passes through a caustic singularity, where the width of the ray tube vanishes. We obtain the JWKB surface wave Green's tensor and derive an explicit expression for the JWKB response to a moment tensor source. The theory allows for slowly varying topography of the Earth's surface and any internal discontinuities, and incorporates the effect of self-gravitation and slight anelasticity on surface wave propagation.
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  • 8
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 103 (1990), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We show how to calculate exact complex eigenfrequencies and eigenfunctions and exact single mode synthetic seismograms for a spherical anelastic earth model. The real frequencies of oscillation of some spheroidal modes differ by 1 to 2 μHz from the corresponding spherical elastic earth eigenfrequencies. The decay rates are generally well approximated by conventional first-order perturbation theory. The complex radial eigenfunctions of the most strongly affected modes differ substantially from the corresponding real elastic eigenfunctions, and this can lead to significant perturbations in the initial phase and amplitude of the associated free oscillations following an earthquake. Most, but not all, of the strongly affected modes have large displacements in the inner core.
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  • 9
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 115 (1993), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: The usual JWKB ray-theoretical description of Love and Rayleigh surface wave propagation on a smooth, laterally heterogeneous earth model breaks down in the vicinity of caustics, near the source and its antipode. In this paper we use Maslov theory to obtain a representation of the wavefield that is valid everywhere, even in the presence of caustics. The surface wave trajectories lie on a 3-D manifold in 4-D phase space (θ, φ, kθ, kφ), where θ is the colatitude, φ is the longitude, and kθ and kφ are the covariant components of the wave vector. There are no caustics in phase space; it is only when the rays are projected onto configuration space (θ, φ), the mixed spaces (kθ, φ) and (θ, kφ), or momentum space (kθ, kφ), that caustics occur. The essential strategy is to employ a mixed-space or momentum-space representation in the vicinity of configuration-space caustics, where the (θ, φ) representation fails. By this means we obtain a uniformly valid Green's tensor and an explicit asymptotic expression for the surface wave response to a moment tensor source.
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  • 10
    Electronic Resource
    Electronic Resource
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 122 (1995), S. 0 
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: A comprehensive study of the Fréchet kernels and group velocities of the Earth's normal modes is conducted based upon the asymptotic eigenfrequency and eigenfunction analyses that we developed previously. Two different approaches, which employ the eigenfunctions or eigenfrequency equations, yield asymptotically equivalent results for the Fréchet kernels and group velocities. The latter approach is considerably simpler, since the need to specify the normal-mode eigenfunctions as well as their radial derivatives is removed, so that the Fréchet kernels and group velocities can be obtained from knowledge of the asymptotic eigenfrequencies only.The asymptotic analyses are discussed for all possible ray parameter regimes and ray-path combinations within a crustless version of the earth model 1066A with two discontinuities: a core-mantle boundary and an inner core boundary. The exact and asymptotic numerical results for the Fréchet kernels and group velocities are compared for such a model. The comparison shows that the asymptotic Fréchet kernels and group velocities are very accurate. The accuracy is better for toroidal modes and relatively high-frequency spheroidal modes.
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