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

标量波和弹性波的逆边值问题：稳定性估计和迭代重建

• 批准号: 1559587
• 负责人: Maarten de Hoop
• 金额: $ 19万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1559587&HistoricalAwards=false
• 关键词: Inverse; Boundary; Value; Problems; Scalar

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.

该项目集中在反向问题上，这使创新的新技术用于解释地震波场中包含的丰富信息。结果将从根本上提高重建高度异质地质媒体的能力，并通过观察数据结构结构。将要开发的非线性方法将产生我们星球内部迄今未知子结构的可能发现，例如裂纹和断层，包括存在流体和地壳膜界面。一方面，浅层和深层结构的更准确的映射和表征将有助于综合的地质和地球物理研究，并可能导致地球动态内部更全面的模型。另一方面，这些方法还将有助于制定监视或确定随时间变化并受益于自然资源管理的策略。结果还将为如何将表面的过程与地球深内部的过程耦合。主要研究者及其同事将对时间谐波和双曲线配方中的地震逆边界价值问题进行全面分析。他们将Cauchy数据和Dirichlet到Neumann地图或Neumann到Dirichlet地图视为数据。不同的配方强调了数据的不同“特征”，并导致稳定恢复的不同条件。主要研究者及其同事将结合使用Helmholtz方程的逆边界值问题，波浪方程及其扩展到描述（时谐）弹性波的系统。他们计划在具有含有综合奇点（接口）和有限平滑度的系数的时谐波弹性波中研究全球唯一性。他们将分析具有部分数据的相关逆图的Lipschitz稳定性条件。这将使团队能够获得稳定性常数的估计值，这将导致系数子空间的层次结构，并通过引入广义变异源条件来开发局部迭代方法的家族。他们还计划开发解决方案估计值，该估计提供了时间谐波和双曲线配方之间的联系，并分析条件，以从高频数据中唯一恢复分段平滑系数。最后，他们将获得一种直接重建边界附近弹性参数（自由表面）的方法，并重新审视使​​用复杂的几何光学解决方案在唯一性理论的证明中使用复杂的几何光学解决方案，并将它们适应到迭代正则化和重建框架中，而数据中没有很低的频率。