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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. 

An elastic map T associates stress with strain in some material. A symmetry of T is a rotation of the material that leaves T unchanged, and the symmetry group of T consists of all such rotations. The symmetry class of T describes the symmetry group but without the orientation information. With an eye toward geophysical applications, Browaeys & Chevrot developed a theory which, for any elastic map T and for each of six symmetry classes Σ, computes the "Σ-percentage" of T. The theory also finds a "hexagonal approximation"—an approximation to T whose symmetry class is at least transverse isotropic. We reexamine their theory and recommend that the Σ-percentages be abandoned. We also recommend that the hexagonal approximations to T be replaced with the closest transverse isotropic maps to T.