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SUMMARY Evidence of seismic anisotropy is widespread within the Earth, including from individual crystals, rocks, borehole measurements, active-source seismic data, and global seismic data. The seismic anisotropy of a material determines how wave speeds vary as a function of propagation direction and polarization, and it is characterized by density and the elastic map, which relates strain and stress in the material. Associated with the elastic map is a symmetric $$6 \times 6$$ matrix, which therefore has 21 parameters. The 21-D space of elastic maps is vast and poses challenges for both theoretical analysis and typical inverse problems. Most estimation approaches using a given set of directional wave speed measurements assume a high-symmetry approximation, typically either in the form of isotropy (2 parameters), vertical transverse isotropy (radial anisotropy: 5 parameters), or horizontal transverse isotropy (azimuthal anisotropy: 6 parameters). We offer a general approach to explore the space of elastic maps by starting with a given elastic map $$\mathbf {T}$$. Using a combined minimization and projection procedure, we calculate the closest $$\Sigma$$-maps to $$\mathbf {T}$$, where $$\Sigma$$ is one of the eight elastic symmetry classes: isotropic, cubic, transverse isotropic, trigonal, tetragonal, orthorhombic, monoclinic and trivial. We apply this approach to 21-parameter elastic maps derived from laboratory measurements of minerals; the measurements include dependencies on pressure, temperature, and composition. We also examine global elasticity models derived from subduction flow modelling. Our approach offers a different perspective on seismic anisotropy and motivates new interpretations, such as for why elasticity varies as a function of pressure, temperature, and composition. The two primary advances of this study are (1) to provide visualization of elastic maps, including along specific pathways through the space of model parameters, and (2) to offer distinct options for reducing the complexity of a given elastic map by providing a higher-symmetry approximation or a lower-anisotropic version. This could contribute to improved imaging and interpretation of Earth structure and dynamics from seismic anisotropy.
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Two Complementary Methods of Inferring Elastic Symmetry
Abstract An elastic map $$\mathbf {T}$$ T describes the strain-stress relation at a particular point $$\mathbf {p}$$ p in some material. A symmetry of $$\mathbf {T}$$ T is a rotation of the material, about $$\mathbf {p}$$ p , that does not change $$\mathbf {T}$$ T . We describe two ways of inferring the group $$\mathcal {S} _{ \mathbf {T} }$$ S T of symmetries of any elastic map $$\mathbf {T}$$ T ; one way is qualitative and visual, the other is quantitative. In the first method, we associate to each $$\mathbf {T}$$ T its “monoclinic distance function” "Equation missing" on the unit sphere. The function "Equation missing" is invariant under all of the symmetries of $$\mathbf {T}$$ T , so the group $$\mathcal {S} _{ \mathbf {T} }$$ S T is seen, approximately, in a contour plot of "Equation missing" . The second method is harder to summarize, but it complements the first by providing an algorithm to compute the symmetry group $$\mathcal {S} _{ \mathbf {T} }$$ S T . In addition to $$\mathcal {S} _{ \mathbf {T} }$$ S T , the algorithm gives a quantitative description of the overall approximate symmetry of $$\mathbf {T}$$ T . Mathematica codes are provided for implementing both the visual and the quantitative approaches.
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- Award ID(s):
- 1829447
- PAR ID:
- 10415814
- Date Published:
- Journal Name:
- Journal of Elasticity
- Volume:
- 150
- Issue:
- 1
- ISSN:
- 0374-3535
- Page Range / eLocation ID:
- 91 to 118
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s10659-022-09898-0
