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.

该项目以反演问题为中心，使创新的新技术能够解释地震波场中包含的丰富信息。研究结果将从根本上提高根据观测数据重建具有结构的高度异质地质介质的能力。将开发的非线性方法将有可能发现地球内部迄今为止未知的子结构，例如裂缝和断层，包括流体的存在和壳幔界面。一方面，更准确地绘制和表征浅层和深层地幔结构将有助于综合地质和地球物理研究，并可能产生更全面的地球动态内部模型。另一方面，这些方法还将有助于制定监测或识别随时间变化的策略，并有利于自然资源管理。研究结果还将为了解地表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对时谐和双曲线公式中的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到诺伊曼映射或诺伊曼到狄利克雷映射视为数据。不同的表述强调数据的不同“特征”，并导致稳定恢复的不同条件。首席研究员和他的同事将结合分析亥姆霍兹方程和波动方程的反边值问题及其对描述（时谐）弹性波系统的扩展。他们计划研究时谐弹性波的全局唯一性，其系数包含共常奇点（界面）和有限的平滑度。他们将使用部分数据分析相关反映射的 Lipschitz 稳定性条件。这将使团队能够获得稳定性常数的估计，从而产生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。他们还计划开发解析估计，提供时谐公式和双曲线公式之间的联系，并分析从高频数据中独特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，并重新审视复杂几何光学解决方案在唯一性定理证明中的使用，并使它们适应迭代正则化和重建的框架，而无需很低的成本。数据中的频率。