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.

这个项目以反问题为中心，使创新的新技术能够解释地震波场中包含的丰富信息。这些结果将从根本上提高从观测数据中重建具有结构的高度非均质地质介质的能力。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亚结构，如裂缝和断层，包括流体和壳幔界面的存在。一方面，对浅部和深部地幔结构进行更准确的测绘和表征将有助于地质和地球物理的综合研究，并可能导致更全面的地球内部动力学模型。另一方面，这些方法还将有助于制定监测或确定随时间变化的战略，并有利于自然资源管理。这些结果还将给我们提供重要的洞察，了解地表的过程如何与地球深处的过程相耦合。首席研究员和他的同事们将对时间调和和双曲公式中的地震反边值问题进行全面的分析。他们认为柯西数据和Dirichlet到Neumann映射或Neumann到Dirichlet映射是数据。不同的公式强调了数据的不同“特征”，导致了稳定恢复的不同条件。主要研究人员和他的同事将联合分析Helmholtz方程和波动方程的反边值问题，以及它们对描述(时间谐和)弹性波的系统的扩展。他们计划研究时间谐和弹性波的全局唯一性，其系数包含正交奇点(界面)和有限的光滑性。他们将分析具有部分数据的相关逆映射的Lipschitz稳定性条件。这将使团队能够获得稳定常数的估计，这将导致系数子空间的分层，并通过引入广义变分源条件来开发一族局部迭代方法。他们还计划开发预解估计，提供时间调和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(自由面)附近弹性参数的方法，并重新讨论在唯一性定理证明中使用复杂几何光学解的方法，并使其适用于迭代正则化和重建的框架，而不需要在数据中使用很低的频率。