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  • 1
    Electronic Resource
    Electronic Resource
    Variational Principles For Surface Wave Propagation On A Laterally Heterogeneous Earth-Iii. Potential Representation (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 112 (1993), S. 0 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb01449.x
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  • 2
    Electronic Resource
    Electronic Resource
    Asymptotic eigenfrequencies of the Earth's normal modes (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 115 (1993), S. 0 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb01490.x
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  • 3
    Electronic Resource
    Electronic Resource
    Variational principles for surface wave propagation on a laterally heterogeneous Earth—I. Time-domain JWKB theory (1992)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 109 (1992), S. 0 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1992.tb00119.x
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  • 4
    Electronic Resource
    Electronic Resource
    Variational principles for surface wave propagation on a laterally heterogeneous Earth—II. Frequency-domain JWKB theory (1992)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 109 (1992), S. 0 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1992.tb00120.x
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  • 5
    Electronic Resource
    Electronic Resource
    Free oscillations of a spherical anelastic earth (1990)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 103 (1990), S. 0 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1990.tb05682.x
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  • 6
    Electronic Resource
    Electronic Resource
    Maslov theory for surface wave propagation on a laterally heterogeneous earth (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 115 (1993), S. 0 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb01203.x
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  • 7
    Electronic Resource
    Electronic Resource
    Surface Wave Caustics (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 114 (1993), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: The geometrical theory of surface wave propagation on a laterally heterogeneous earth model diverges at caustics, where neighbouring rays cross. the caustics on a spherical earth are degenerate focal points at the source and its antipode; lateral heterogeneity removes this degeneracy and transforms the caustics into multiply cusped and folded curves. We investigate the geometric nature of these antipodal and source caustics using both linear ray-perturbation theory and exact ray tracing. the regions occupied by the R1-R2 and R2-R3 caustics extend as far as 20° from the antipode and 30° from source, respectively, even on a relatively smooth earth model such as M84A.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb03920.x
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  • 8
    Electronic Resource
    Electronic Resource
    Summation of the Born series for the normal modes of the Earth (1990)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 100 (1990), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We derive an explicit formula for the response of a laterally heterogeneous, self-gravitating, rotating, dissipative, and physically dispersive earth to an earthquake. The long-period spectrum at a single station is given by u(ω) =RH0 [V(ω) +ωW−ω2T]−1·S0(ω) where R0 and S0(ω) are the unperturbed source and receiver vectors and H denotes the Hermitian transpose. The quantities V(ω), W, and T are the potential energy, Coriolis, and relative kinetic energy matrices, and V(ω) +ωW−ω2T is the Lagrangian matrix whose eigenvalues are the perturbed complex eigenfrequencies. This simple result is obtained by formal summation of the Born series.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1990.tb00704.x
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  • 9
    Electronic Resource
    Electronic Resource
    JWKB surface-wave seismograms on a laterally heterogeneous earth (1994)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 119 (1994), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: A numerically efficient implementation of JWKB theory is developed to calculate synthetic surface-wave seismograms on a smooth, laterally heterogeneous earth model. We illustrate the method by applying it to model SH12/WM13; the phase and amplitude anomalies of long-period, fundamental-mode Love and Rayleigh waves are compared with the corresponding results obtained using the great-circle approximation and first-order ray-perturbation theory, with the perturbations to the local eigenfunctions ignored. As expected, the most serious discrepancies between these two approximations and full JWKB theory occur for paths with large transverse phase-velocity gradients, which lead to significant deviations away from the source-receiver great circle. The great-circle approximation consistently under-estimates the phase of the first-arriving G1 Love and R1 Rayleigh waves, in accordance with Fermat's principle of least time. The global average Fermat bias is greater for Love than for Rayleigh waves; this may be partly responsible for the observed Love-Rayleigh discrepancy currently attributed to upper mantle transverse isotropy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1994.tb00130.x
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  • 10
    Electronic Resource
    Electronic Resource
    Surface Wave Propagation In A Slowly Varying Anisotropic Waveguide (1993)
    Oxford, UK : Blackwell Publishing Ltd
    Geophysical journal international 113 (1993), S. 0 
    Details
    ISSN: 1365-246X
    Source: Blackwell Publishing Journal Backfiles 1879-2005
    Topics: Geosciences
    Notes: We present a JWKB theory which describes the propagation of seismic surface waves in a laterally heterogeneous, anisotropic waveguide. We introduce a local dispersion relation and local vertical eigenfunctions which depend explicitly on the direction of the local wavevector as a consequence of the anisotropy. the variation of amplitude along a surface wave ray path is determined by a conservation law for the surface wave energy. Apart from the usual dynamical phase, which is the integral of the local wavevector along a ray path, there is an additional variation in phase in a general anisotropic waveguide. This additional phase, which is an analogue of the Berry phase in adiabatic quantum mechanics, vanishes in a waveguide with a local vertical two-fold symmetry axis or a local horizontal mirror plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1111/j.1365-246X.1993.tb02542.x
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