Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

Bóna, Andrej; Bucataru, Ioan; Slawiński, Michael A.

doi:10.1007/s10659-007-9099-z

Cited by 46 publications

(34 citation statements)

References 20 publications

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“…A realistic representation of propagating wavefields is key to creating accurate subsurface models in global and exploration seismology. Elastic anisotropy, which can be classified and decomposed based upon the symmetries present in the anisotropic media (Bóna et al, ; Browaeys & Chevrot, ), strongly influences the propagation of wavefields throughout the Earth's interior. In global seismology, inversions of the anisotropy of the inner core (Creager, ; Tromp, ) and some regions of the Earth's mantle (e.g., Bozdăg et al, ; Long & Becker, ) have been used to construct more realistic global models for a better understanding of the physics of the Earth.…”

Section: Introductionmentioning

confidence: 99%

Scattering Radiation Pattern Atlas: What Anisotropic Elastic Properties Can Body Waves Resolve?

Kazei

Alkhalifah

2019

JGR Solid Earth

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Full‐waveform inversion (FWI) optimizes the subsurface properties of geophysical Earth models in such a way that the modeled data, based on these subsurface properties, match the observed data. The anisotropic properties, whether monoclinic, orthorhombic, triclinic, or vertical transversally isotropic (VTI), of the subsurface, be it a fractured reservoir or the core‐mantle boundary, are necessary to describe the observed wave phenomena. There are no principal limitations on the complexity of the anisotropy that can be inverted using FWI. However, the question remains—what kind of anisotropic descriptions of the elastic properties of the Earth can or cannot be inverted reliably from seismic waveforms? We reveal the resolution that can be achieved through reconstructions of each elastic parameter by building vertical resolution patterns from the scattering radiation patterns of body waves. A visual analysis of these patterns indicates “trade‐offs,” that is, perturbations of parameters that have the same reflection‐based scattering patterns as other perturbations. Each trade‐off leads to an apparent ambiguity in the inversion, which must be addressed by additional assumptions, constraints, or regularizations. For orthorhombic media, we find the exact trade‐offs that exist between parameters. Our parameterization isolates the VTI parameters, and therefore, we also obtain trade‐offs for every scattering mode for VTI media. We discover that only monoclinic parameters are recoverable from the first‐order scattering of monotypic waves. We summarize the trade‐offs in tables for easy reference. This paper is intended to be useful for researchers setting up anisotropic FWI problems, and interpreting or controlling the quality of such inversions.

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Section: Introductionmentioning

confidence: 99%

Scattering Radiation Pattern Atlas: What Anisotropic Elastic Properties Can Body Waves Resolve?

Kazei

Alkhalifah

2019

JGR Solid Earth

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“…Using results with parameters , we obtain

\begin{matrix} true \\ left c_{1111} = (() 5.8472, 5.8472, 5.8472, 5.8472) and \\ left c_{2323} = (() 0.2299, 4.3506, 4.3506, 4.3506); \end{matrix}

the Backus average parameters, following expressions (10), are

\begin{matrix} true \\ right c_{1111}^{\bar{TI}} & = & left 3.6692, c_{1122}^{\bar{TI}} = - 2.9717, c_{3333}^{\bar{TI}} = 5.8472, \\ right c_{1133}^{\bar{TI}} & = & left - 0.7936, c_{2323}^{\bar{TI}} = 0.7936, c_{1212}^{\bar{TI}} = 3.3204 . \end{matrix}

The eigenvalues of tensor with values are

λ_{1} = λ_{2} = 6.6409

λ_{3} = 6.0812

λ_{4} = λ_{5} = 1.5873

λ_{6} = 0.4635

, which belong to a transversely isotropic tensor (Bóna, Bucataru and Slawinski ); since they are positive, the stability condition of the average is satisfied. Also,

Disc false(normalΔ false) = 1.9883 \times 10^{- 12}

, which can be considered zero, as required.…”

Section: Christoffel Rootsmentioning

confidence: 97%

“…The eigenvalues of tensor (2) with values (21) are λ 1 = λ 2 = 6.6409, λ 3 = 6.0812, λ 4 = λ 5 = 1.5873, λ 6 = 0.4635, which belong to a transversely isotropic tensor (Bóna, Bucataru and Slawinski 2007a); since they are positive, the stability condition of the average is satisfied. Also, Disc( ) = 1.9883 × 10 −12 , which can be considered zero, as required.…”

Section: Numerical Examplementioning

confidence: 99%

On Christoffel roots for nondetached slowness surfaces

Bos

Slawiński

Stanoev

2019

Geophysical Prospecting

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The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine the Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media. If the qP slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the qP slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the qP and qSV slowness surfaces are not smooth.

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“…The eigensystem coordinate‐free characterization of the material symmetries 30 states that a material symmetry class can be defined by both the multiplicities of the eigenvalues and constraints on the related eigenspaces. Specifically, it can be shown for instance that an isotropic (resp.…”

Section: Definition Of Constraints With Respect To Materials Symmetrymentioning

confidence: 99%

“…Consequently, we consider the projection onto the set ℰ TI of all the elasticity tensors exhibiting transverse isotropy with respect to e 3 , defined with respect to the Euclidean distance d E (see 29). Following 30 and Section 2.8, the mean distance D TI can be specified, for a given overall level of statistical fluctuation, by setting an appropriate value of parameter τ. The plot of function τ↦D TI (60, (τ, τ, 0, τ, τ, 0)), obtained using the Riemannian distance, is shown in Figure 4.…”

Section: Definition Of Constraints With Respect To Materials Symmetrymentioning

confidence: 99%

Non‐Gaussian positive‐definite matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries

Guilleminot¹,

Soize²

2011

Numerical Meth Engineering

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SUMMARYThis paper is devoted to the construction of a class of prior stochastic models for non-Gaussian positivedefinite matrix-valued random fields. The proposed class allows the variances of selected random eigenvalues to be specified and exhibits a larger number of parameters than the other classes previously derived within a nonparametric framework. Having recourse to a particular characterization of material symmetry classes, we then propose a mechanical interpretation of the constraints and subsequently show that the probabilistic model may allow prescribing higher statistical fluctuations in given directions. Such stochastic fields turn out to be especially suitable for experimental identification under material symmetry uncertainties, as well as for the development of computational multi-scale approaches where the randomness induced by fine-scale features may be taken into account. We further present a possible strategy for inverse identification, relying on the sequential solving of least-square optimization problems. An application is finally provided.

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Coordinate-free Characterization of the Symmetry Classes of Elasticity Tensors

Abstract: We formulate coordinate-free conditions for identifying all the symmetry classes of the elasticity tensor and prove that these conditions are both necessary and sufficient. Also, we construct a natural coordinate system of this tensor without the a priory knowledge of the symmetry axes.

Cited by 46 publications

References 20 publications

Scattering Radiation Pattern Atlas: What Anisotropic Elastic Properties Can Body Waves Resolve?

Scattering Radiation Pattern Atlas: What Anisotropic Elastic Properties Can Body Waves Resolve?

On Christoffel roots for nondetached slowness surfaces

Non‐Gaussian positive‐definite matrix‐valued random fields with constrained eigenvalues: Application to random elasticity tensors with uncertain material symmetries

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