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

该项目以逆问题为中心，为解释地震波场中包含的丰富信息提供了创新的新技术。研究结果将从根本上提高利用观测资料重建高度非均质地质介质的能力。将要发展的非线性方法将有可能发现我们星球内部迄今未知的子结构，如裂缝和断层，包括流体和地壳-地幔界面的存在。一方面，对浅层和深层地幔结构更精确的制图和表征将促进地质和地球物理综合研究，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监测或查明随时间变化的战略，并有利于自然资源管理。这些结果还将对地表的过程如何与地球深处的过程耦合提供重要的见解。首席研究员和他的同事将在时间调和和双曲公式中对地震反边值问题进行全面分析。他们将柯西数据和狄利克雷-诺伊曼图或诺伊曼-狄利克雷图作为数据。不同的配方强调不同的数据“特征”，导致稳定回收的条件不同。首席研究员和他的同事将共同分析亥姆霍兹方程和波动方程的反边值问题，以及它们在描述（时谐）弹性波的系统中的扩展。他们计划在系数包含正交奇点（界面）和有限平滑度的时谐弹性波的情况下研究全局唯一性。他们将用部分数据分析相关逆映射的Lipschitz稳定性条件。这将使团队能够获得稳定性常数的估计，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开发一系列局部迭代方法。他们还计划开发解决方案，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恢复分段平滑系数的唯一条件。最后，他们将获得一种在边界（自由曲面）附近直接重建弹性参数的方法，并重新审视在唯一性定理证明中使用复杂几何光学解，并使其适应迭代正则化和重构的框架，而数据中没有非常低的频率。

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.