Ray Tracing On A Heterogeneous Sphere By Lie Series

Virieux, Jean; Ekström, Göran

doi:10.1111/j.1365-246x.1991.tb02491.x

Cited by 4 publications

(2 citation statements)

References 35 publications

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“…Abraham & Marsden 1978, section 3.2). This property suggests a potentially interesting link with the work of Virieux & Ekström (1991) who employ the method of Lie series to generate canonical transformations which simplify the solution of the surface wave ray tracing equations in laterally heterogeneous earth models. Though we have not investigated the relation between these two methods in any detail, it may prove fruitful to do so in later work.…”

Section: Asymptotic Ray Theorymentioning

confidence: 96%

Particle relabelling transformations in elastodynamics

Al-Attar¹,

Crawford²

2016

Geophys. J. Int.

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S U M M A R YThe motion of a self-gravitating hyperelastic body is described through a time-dependent mapping from a reference body into physical space, and its material properties are determined by a referential density and strain-energy function defined relative to the reference body. Points within the reference body do not have a direct physical meaning, but instead act as particle labels that could be assigned in different ways. We use Hamilton's principle to determine how the referential density and strain-energy functions transform when the particle labels are changed, and describe an associated 'particle relabelling symmetry'. We apply these results to linearized elastic wave propagation and discuss their implications for seismological inverse problems. In particular, we show that the effects of boundary topography on elastic wave propagation can be mapped exactly into volumetric heterogeneity while preserving the form of the equations of motion. Several numerical calculations are presented to illustrate our results.

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Section: Asymptotic Ray Theorymentioning

confidence: 96%

Particle relabelling transformations in elastodynamics

Al-Attar¹,

Crawford²

2016

Geophys. J. Int.

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“…Let us define the Harniltonian (Burridge 1976;Virieux & Ekstrom 1991;Tromp & Dahlen 1992) H ( 8, @, k,, k,) = i [ k i + k ; sinP28 -k2( 8, @)I.…”

Section: Ray Tracingmentioning

confidence: 99%

Maslov theory for surface wave propagation on a laterally heterogeneous earth

Tromp

Dahlen

1993

Geophysical Journal International

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S U M M A R YT h e usual J W K B ray-theoretical description of Love a n d 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 t o obtain a representation of the wavefield that is valid everywhere, even in the presence of caustics. T h e surface wave trajectories lie o n a 3-D manifold in 4-D phase space (0, @, k e , k,), where 0 is the colatitude, @ is the longitude, and k , and k, 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 (0, @), the mixed spaces (k,, @) and (0, k @ ) , or momentum space (k,, k,), that caustics occur. T h e essential strategy is t o employ a mixed-space o r momentum-space representation in the vicinity of configuration-space caustics, where the (0, @) 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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Variational principles for surface wave propagation on a laterally heterogeneous Earth--I. Time-domain JWKB theory

Tromp

Dahlen

1992

Geophysical Journal International

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S U M M A R YW e present a J W K B theory that describes t h e 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. T h e local dispersion relations determine the surface wave trajectories, and the energy equations determine the surface wave amplitudes. T h e amplitude of a surface wavegroup varies in time as a result of both dispersion and geometrical spreading. T h e 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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Ray Tracing On A Heterogeneous Sphere By Lie Series

Cited by 4 publications

References 35 publications

Particle relabelling transformations in elastodynamics

Particle relabelling transformations in elastodynamics

Maslov theory for surface wave propagation on a laterally heterogeneous earth

Variational principles for surface wave propagation on a laterally heterogeneous Earth--I. Time-domain JWKB theory

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