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越许多尺度的逆问题。这些反问题与描述弹性波和弹性静力学的偏微分方程系统有关。他们在这些问题上的主要进展涉及通过考虑散射、各向异性和非线性弹性来达到真实的地球材料特性及其复杂的非均匀性。应用程序的目标是严格的重建，分层映射地球的内部，局部和可能的全球，尊重这些材料的属性，结合位错和断层形状编码的同震变形。研究人员利用新的传感器设计和数据采集的重要发展。区分所观察到的分层数据的复杂性，预计将与不同的散射制度的功能，调查区分不同类型的非线性反问题。这项研究有望导致创新技术，解释地震和大地测量（全球定位系统）数据中包含的丰富信息，一方面跨越尺度，另一方面在分析支撑现代数据科学的逆问题方面开辟新的方向。这些研究有益于自然资源管理，包括水力压裂和地热能，危险分析，此外，还为用很少的传感器进行行星探索提供了可能的方法。它们还为进一步深入了解地表过程如何与地球内部深处的过程相耦合提供了关键。该研究计划为参与的学生提供了独特的跨学科教育体验，使他们对新技术和现实生活影响的重要性有了更广泛的认识。 研究者和他的合作者开发了一个地震学和大地测量学的复合分析的反问题。根据数据的复杂性，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，以及在相界和相界附近的（瑞利和斯通利）波导耦合（2）研究黎曼几何中秩4张量上局部混合测地线射线变换的几何反问题，Finsler几何中的（破）测地线射线变换，然后是边界刚性，并与可见定向测地线球以及与微震活动性数据的几何反问题;（3）分析具有分片光滑刚度张量的双曲型反边值问题（具有相关联的接口或域分区）、部分数据和使用用于基于优化的重建的不平衡的、复杂的最优传输的改进的稳定性估计;（4）研究由大地测量边界数据恢复非均匀位错和断层形状的唯一性和条件稳定性;（5）利用成对拉格朗日分布和Eschhartz估计，建立并分析了存在不连续面时沉积岩非线性弹性力学的反问题。这些逆问题之间的隐含联系在它们探测一个星球时发挥作用，同时有助于更深入地了解通过现代数据输入可获得的呈指数增长的数据量中的信息内容。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.