Seismology- and Geodesy-Based Inverse Problems Crossing Scales, with Scattering, Anisotropy and Nonlinear Elasticity

研究者和他的合作者研究地震学和大地测量学中跨多个尺度的逆问题。这些反问题与描述弹性波和弹性静力学的偏微分方程组有关。他们在这些问题上的关键进展在于通过考虑散射、各向异性和非线性弹性来达到真实地球材料的性质及其复杂的非均质性。应用目标是严格的重建，在局部和可能的全球范围内分层绘制地球内部地图，尊重这些材料特性，结合同震变形中编码的位错和断层形状。研究者利用新的传感器设计和数据采集的重要发展。研究人员区分了不同类型的非线性逆问题，区分了代表观察到的分层数据复杂性的特征，预计与不同的散射制度有关。该研究一方面有望带来解释地震和测地线（全球定位系统）跨尺度数据中包含的丰富信息的创新技术，另一方面也有望带来支撑现代数据科学的逆问题分析的新方向。这些研究有利于自然资源管理，包括水力压裂法和地热能，危害分析，此外，还为很少传感器的行星探测提供了可能的方法。它们还为进一步了解地表的过程如何与地球深处的过程耦合提供了重要的关键。该研究项目为参与的学生提供了独特的跨学科教育体验，使他们对新技术的重要性和现实生活中的影响有了更广泛的认识。研究者和他的合作者开发了一种基于逆问题的地震学和大地测量学的复合分析。根据分层数据复杂性，他们(1)分析球对称行星的光谱刚性，然后通过微扰和半经典分析包括角度变化，以及（Rayleigh和Stoneley）波导耦合在相位边界和附近的逆问题；(2)研究了黎曼几何中局部混合测地线射线在秩4张量上的变换、芬斯勒几何中（破碎）测地线射线在边界刚度上的变换、可见定向测地线球的几何逆问题以及微震活动数据的几何逆问题；(3)用分段光滑刚度张量（带相关界面或域分区）、部分数据和改进的稳定性估计分析双曲型反边值问题，利用不平衡、复杂的最优传输进行优化重建；(4)研究了从大地测量边界数据中恢复非均质位错和断层形状的唯一性和条件稳定性；(5)利用配对拉格朗日分布和Strichartz估计，发展和分析存在不连续面时沉积岩非线性弹性的反问题。当它们探测一个星球时，这些反向问题之间的隐含联系开始发挥作用，同时有助于更深入地理解通过现代数据中心提供的指数级增长的数据量中的信息内容。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The investigator and his collaborators study inverse problems in seismology and geodesy crossing many scales. These inverse problems are associated with systems of partial differential equations describing elastic waves and elastostatics. Their key advances in these problems pertain to reaching real earth material properties and their complex heterogeneities through the consideration of scattering, anisotropy and nonlinear elasticity. The application goal is rigorous reconstruction, hierarchically mapping Earth's interior, locally and possibly globally, honoring these material properties, in conjunction with dislocations and fault shapes encoded in co-seismic deformation. The investigator exploits important developments in novel sensor design and data acquisition. Differentiating the features representing the observed hierarchical data complexity, expected to be associated with different scattering regimes, the investigator distinguishes different types of nonlinear inverse problems. The research promises to lead to innovative technologies for interpreting the rich information contained in seismic and geodesic (Global Positioning System) data crossing scales on the one hand, and new directions in the analysis of inverse problems underpinning modern data science on the other hand. The studies benefit natural resources management including hydrofracking and geothermal energy, hazard analysis and, moreover, provide possible approaches for planetary exploration with very few sensors. They also provide keys for important further insight on how processes at the surface are coupled to processes in Earth's deep interior. The research program offers a unique interdisciplinary educational experience for the students involved giving them a much broader appreciation of the importance of novel techniques and real-life implications. The investigator and his collaborators develop a composite analysis of seismology and geodesy based inverse problems. Following the hierarchical data complexity, they (1) analyze spectral rigidity of spherically symmetric planets, and then include angular variations through perturbation and semiclassical analysis, as well as inverse problems for (Rayleigh and Stoneley) waveguide coupling at and near phase boundaries; (2) study geometric inverse problems for the local mixed geodesic ray transforms on rank-4 tensors in Riemannian geometry, (broken) geodesic ray transforms in Finsler geometry and then boundary rigidity, and with visible orientable geodesic spheres as well as the geometric inverse problem with microseismicity data; (3) analyze hyperbolic inverse boundary value problems with piecewise smooth stiffness tensors (with associated interfaces or domain partitions), partial data and improved stability estimates using unbalanced, complex optimal transport for optimization-based reconstruction; (4) study uniqueness and conditional stability for the recovery of a heterogeneous dislocation and fault shape from geodesy boundary data; and (5) develop and analyze inverse problems in nonlinear elasticity in sedimentary rocks in the presence of discontinuities using paired Lagrangian distributions and Strichartz estimates. The implicit connections between these inverse problems come into play as they probe one planet, while contributing to a deeper understanding of information content in the exponentially growing data volumes that are available through modern data enters.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.