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

这位研究人员和他的合作者研究了地震学和大地测量学中跨越多个尺度的反问题。这些逆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主要进展涉及通过考虑散射、各向异性和非线性弹性来获得真实地球材料的性质及其复杂的非均质性。应用程序的目标是严格重建，分层绘制地球内部、局部和可能的全球地图，尊重这些材料属性，并结合在同震变形中编码的位错和断层形状。研究人员开发了新型传感器设计和数据采集方面的重要发展。通过区分代表观测到的分层数据复杂性的特征，预计这些特征与不同的散射机制相关联，研究者区分不同类型的非线性反问题。这项研究一方面有望带来解释地震和测地线(全球定位系统)数据中包含的丰富信息的创新技术，另一方面将在支持现代数据科学的反问题分析方面产生新的方向。这些研究有利于包括水力压裂和地热能在内的自然资源管理、危险分析，而且还为使用极少传感器进行行星探测提供了可能的途径。它们还为进一步深入了解地球表面的过程如何与地球内部的过程相耦合提供了关键。该研究项目为参与研究的学生提供了一种独特的跨学科教育体验，让他们更广泛地了解新技术的重要性和现实生活中的影响。这位研究人员和他的合作者开发了一个基于地震学和大地测量学的反问题的综合分析。根据层次数据的复杂性，(1)分析了球对称行星的谱刚性，然后通过摄动和半经典分析包括角度变化，以及相边界上和附近的(Rayleigh和Stoneley)波导耦合的反问题，(2)研究了黎曼几何中阶-4张量的局部混合测地线变换的几何反问题，Finsler几何中的(断)测地线变换和边界刚性，以及可见可定向测地球面的几何反问题以及微震活动数据的几何反问题；(3)分析具有分段光滑刚度张量(具有相关界面或区域划分)、部分数据和改进的稳定性估计的双曲型反边值问题；(4)研究从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Strichartz估计，发展和分析存在不连续面的沉积岩中的非线性弹性反问题。这些反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理解通过现代数据输入获得的指数级增长的数据量中的信息内容。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.