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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. 

Abstract With the rise of data volume and computing power, seismological research requires more advanced skills in data processing, numerical methods, and parallel computing. We present the experience of conducting training workshops in various forms of delivery to support the adoption of large-scale high-performance computing (HPC) and cloud computing, advancing seismological research. The seismological foci were on earthquake source parameter estimation in catalogs, forward and adjoint wavefield simulations in 2D and 3D at local, regional, and global scales, earthquake dynamics, ambient noise seismology, and machine learning. This contribution describes the series of workshops delivered as part of research projects, the learning outcomes for participants, and lessons learned by the instructors. Our curriculum was grounded on open and reproducible science, large-scale scientific computing and data mining, and computing infrastructure (access and usage) for HPC and the cloud. We also describe the types of teaching materials that have proven beneficial to the instruction and the sustainability of the program. We propose guidelines to deliver future workshops on these topics. 

ABSTRACT Seismic tomography is the most abundant source of information about the internal structure of the Earth at scales ranging from a few meters to thousands of kilometers. It constrains the properties of active volcanoes, earthquake fault zones, deep reservoirs and storage sites, glaciers and ice sheets, or the entire globe. It contributes to outstanding societal problems related to natural hazards, resource exploration, underground storage, and many more. The recent advances in seismic tomography are being translated to nondestructive testing, medical ultrasound, and helioseismology. Nearly 50 yr after its first successful applications, this article offers a snapshot of modern seismic tomography. Focused on major challenges and particularly promising research directions, it is intended to guide both Earth science professionals and early-career scientists. The individual contributions by the coauthors provide diverse perspectives on topics that may at first seem disconnected but are closely tied together by a few coherent threads: multiparameter inversion for properties related to dynamic processes, data quality, and geographic coverage, uncertainty quantification that is useful for geologic interpretation, new formulations of tomographic inverse problems that address concrete geologic questions more directly, and the presentation and quantitative comparison of tomographic models. It remains to be seen which of these problems will be considered solved, solved to some extent, or practically unsolvable over the next decade. 

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. 

The elastic map, or generalized Hooke’s Law, associates stress with strain in an elastic material. A symmetry of the elastic map is a reorientation of the material that does not change the map. We treat the topic of elastic symmetry conceptually and pictorially. The elastic map is assumed to be linear, and we study it using standard notions from linear algebra—not tensor algebra. We depict strain and stress using the “beachballs” familiar to seismologists. The elastic map, whose inputs and outputs are strains and stresses, is in turn depicted using beachballs. We are able to infer the symmetries for most elastic maps, sometimes just by inspection of their beachball depictions. Many of our results will be familiar, but our versions are simpler and more transparent than their counterparts in the literature. 

Abstract We use earthquake‐based adjoint tomography to invert for three‐dimensional structure of the North Island, New Zealand, and the adjacent Hikurangi subduction zone. The study area, having a shallow depth to the plate interface below the North Island, offers a rare opportunity for imaging material properties at an active subduction zone using land‐based measurements. Starting from an initial model derived using ray tomography, we perform iterative model updates using spectral element and adjoint simulations to fit waveforms with periods ranging from 4–30 s. We perform 28 model updates using an L‐BFGS optimization algorithm, improving data fit and introducingP‐ andS‐wave velocity changes of up to ±30%. Resolution analysis using point spread functions show that our measurements are most sensitive to heterogeneities in the upper 30 km. The most striking velocity changes coincide with areas related to the active Hikurangi subduction zone. Lateral velocity structures in the upper 5 km correlate well with New Zealand geology. The inversion reveals increased along‐strike heterogeneity on the margin. In Cook Strait we observe a low‐velocity zone interpreted as deep sedimentary basins. In the central North Island, low‐velocity anomalies are linked to surface geology, and we relate velocity structures at depth to crustal magmatic activity below the Taupō Volcanic Zone. Our velocity model provides more accurate synthetic seismograms with respect to the initial model, better constrains small (50 km), shallow (15 km) and near‐offshore velocity structures, and improves our understanding of volcanic and tectonic structures related to the active Hikurangi subduction zone. 

Abstract The crustal structure in south‐central Alaska has been influenced by terrane accretion, flat slab subduction, and a modern strike‐slip fault system. Within the active subduction system, the presence of the Denali Volcanic Gap (DVG), a ∼400 km region separating the active volcanism of the Aleutian Arc to the west and the Wrangell volcanoes to the east, remains enigmatic. To better understand the regional tectonics and the nature of the volcanic gap, we deployed a month‐long north‐south linear geophone array of 306 stations with an interstation distance of 1 km across the Alaska Range. By calculating multi‐component noise cross‐correlation and jointly inverting Rayleigh wave phase velocity and ellipticity across the array, we construct a 2‐D shear wave velocity model along the transect down to ∼16 km depth. In the shallow crust, we observe low‐velocity structures associated with sedimentary basins and image the Denali fault as a narrow localized low‐velocity anomaly extending to at least 12 km depth. About 12 km, below the fold and thrust fault system in the northern flank of the Alaska Range, we observe a prominent low‐velocity zone with more than 15% velocity reduction. Our velocity model is consistent with known geological features and reveals a previously unknown low‐velocity zone that we interpret as a magmatic feature. Based on this feature's spatial relationship to the Buzzard Creek and Jumbo Dome volcanoes and the location above the subducting Pacific Plate, we interpret the low‐velocity zone as a previously unknown subduction‐related crustal magma reservoir located beneath the DVG. 

Abstract Cook Inlet fore‐arc basin in south‐central Alaska is a large, deep (7.6 km) sedimentary basin with the Anchorage metropolitan region on its margins. From 2015 to 2017, a set of 28 broadband seismic stations was deployed in the region as part of the Southern Alaska Lithosphere and Mantle Observation Network (SALMON) project. The SALMON stations, which also cover the remote western portion of Cook Inlet basin and the back‐arc region, form the basis for our observational study of the seismic response of Cook Inlet basin. We quantify the influence of Cook Inlet basin on the seismic wavefield using three data sets: (1) ambient‐noise amplitudes of 18 basin stations relative to a nonbasin reference station, (2) earthquake ground‐motion metrics for 34 crustal and intraslab earthquakes, and (3) spectral ratios (SRs) between basin stations and nonbasin stations for the same earthquakes. For all analyses, we examine how quantities vary with the frequency content of the seismic signal and with the basin depth at each station. Seismic waves from earthquakes and from ambient noise are amplified within Cook Inlet basin. At low frequencies (0.1–0.5 Hz), ambient‐noise ratios and earthquake SRs are in a general agreement with power amplification of 6–14 dB, corresponding to amplitude amplification factors of 2.0–5.0. At high frequencies (0.5–4.0 Hz), the basin amplifies the earthquake wavefield by similar factors. Our results indicate stronger amplification for the deeper basin stations such as near Nikiski on the Kenai Peninsula and weaker amplification near the margins of the basin. Future work devoted to 3D wavefield simulations and treatment of source and propagation effects should improve the characterization of the frequency‐dependent response of Cook Inlet basin to recorded and scenario earthquakes in the region.