Inverse Boundary Value Problems For Scalar and Elastic Waves: Stability Estimates and Iterative Reconstruction

该项目的核心是反问题，使创新的新技术，解释地震波场中包含的丰富信息。该成果将从根本上提高从观测数据重建具有结构的高度非均匀地质介质的能力。将开发的非线性方法将可能发现我们星球内部迄今未知的子结构，如裂缝和断层，包括流体和壳幔界面的存在。一方面，更准确地绘制和描述浅地幔和深地幔结构将有助于综合地质和地球物理研究，并可能导致更全面的地球动态内部模型。另一方面，这些方法也将有助于制定战略，监测或查明随着时间的推移而发生的变化，并有利于自然资源管理。研究结果还将提供重要的洞察力，了解地表的过程如何与地球内部深处的过程相耦合。首席研究员和他的同事将开发一个全面的分析地震逆边值问题的时谐和双曲公式。他们认为柯西数据和狄利克雷到诺依曼映射或诺依曼到狄利克雷映射作为数据。不同的公式强调数据的不同“特征”，并导致稳定恢复的不同条件。主要研究者和他的同事将结合分析亥姆霍兹方程和波动方程的逆边值问题及其扩展到描述（时间谐波）弹性波的系统。他们计划研究全球唯一性的情况下，时间谐波弹性波的系数包含余法奇点（接口）和有限的光滑度。他们将分析条件的Lipschitz稳定性的相关逆映射的部分数据。这将使该团队能够获得稳定常数的估计，从而导致系数子空间的层次结构，并通过引入广义变分源条件来开发一系列局部迭代方法。他们还计划开发预解估计，提供时间谐波和双曲公式之间的联系，并分析从高频数据中唯一恢复分段平滑系数的条件。最后，他们将获得一种直接重建边界（自由表面）附近弹性参数的方法，并重新使用复杂的几何光学解决方案来证明唯一性定理，并使其适应迭代正则化和重建的框架，而不会在数据中出现非常低的频率。

This project centers on inverse problems that enable innovative new technologies for interpreting the rich information contained in seismic wavefields. The results will fundamentally improve the ability to reconstruct highly heterogeneous geological media with structure from observational data. The nonlinear approaches that will be developed will yield possible discoveries of hitherto unknown substructures in our planet's interior, such as cracks and faults including the presence of fluids and the crust-mantle interface. On the one hand, more accurate mapping and characterization of shallow and deep mantle structures will facilitate integrated geological and geophysical studies, and may lead to more comprehensive models of Earth's dynamic interior. On the other hand, these methods will also aid in developing strategies for monitoring or identifying changes over time and benefit natural resources management. The results will also give important insight in how processes at the surface are coupled to processes in Earth's deep interior. The principal investigator and his colleagues will develop a comprehensive analysis of the seismic inverse boundary value problems in the time-harmonic and hyperbolic formulations. They consider Cauchy data and the Dirichlet-to-Neumann map or the Neumann-to-Dirichlet map as the data. The different formulations emphasize different 'features' of the data and lead to different conditions for stable recovery. The principal investigator and his colleagues will analyze in conjunction the inverse boundary value problems for the Helmholtz equation and the wave equation and their extensions to systems describing (time-harmonic) elastic waves. They plan to study global uniqueness in the case of time-harmonic elastic waves with coefficients containing conormal singularities (interfaces) and of limited smoothness. They will analyze conditions for Lipschitz stability of the relevant inverse maps with partial data. This will enable the team to obtain estimates for the stability constants, which leads to hierarchies of subspaces of coefficients, and develop a family of local iterative methods via the introduction of generalized variational source conditions. They also plan to develop resolvent estimates which provide a connection between the time-harmonic and hyperbolic formulations and analyze conditions for the unique recovery of piecewise smooth coefficients from high-frequency data. Finally, they will obtain a method of direct reconstruction of elastic parameters near the boundary (a free surface), and revisit the use of complex geometrical optics solutions in proofs of uniqueness theorems and adapt them to a framework of iterative regularization and reconstruction without very low frequencies in the data.