时域波动方程反散射问题的聚焦成像算法研究

The time domain inverse scattering problems for acoustic and elastic waves arise in various application areas. The study of numerical methods for these problems is an area of major importance in modern computational physics. This project is devoted to an inverse scattering problem in the time domain, i.e., reconstructing the shape of the unknown scatterer from time-dependent scattered waves. Considerable achievements have been made towards the classical inverse scattering problems in the frequency domain, in particular, a significant concern has been devoted to fast imaging methods during the last decades. However, the investigation into the numerical approaches in the time domain is a relatively new research topic. We mainly consider the time domain "total focusing"-type direct imaging method in this project, including: 1. We will try to prove the “boudary focusing” properties of the indicator functional in the acoustic case. 2. For the case of elastic wave, we will study the new schemes of the indicator which are based on separation of wave fields. The effectiveness will be shown both theoretically and numerically. We believe that the expected results of this project would not only enrich the study of time domain inverse scattering problems, but also serve as a useful motivation for other fast methods for time-dependent inverse problems. The results will have practical applications as well.

时域声波和弹性波反散射问题具有广泛的应用背景, 有关的数值算法是现代计算物理中的重要研究领域。本项目研究时域反散射问题，即利用与时间相关的散射波场数据，重构未知散射体的位置和形状。经典频率域反散射问题的研究已经取得了大量重要成果, 特别是快速成像算法近年来得到了极大关注, 但关于时间域反散射问题的快速算法的研究则是一个较新的课题, 有很多科学问题值得探讨。本项目的研究内容是时域“全聚焦”型直接成像算法的理论分析和数值实现, 包括：1. 针对声波情形, 给出算法中指示函数的“边界聚焦”性质的数学理论分析; 2. 针对弹性波情形, 采用波场分离技术设计构造更有效的指示函数, 并通过理论和数值实例验证其有效性。本项目的预期成果, 不仅能进一步丰富时域形状反演问题快速算法的数学理论, 同时也将对研究其它时间相关的反问题提供有益参考, 并且具有实际的应用价值。

时域和频域反散射问题的研究由实际应用问题驱动，相关研究具有理论意义和潜在应用价值。本项目围绕时域声波和弹性波反散射问题的全聚焦方法，开展了几类时域和频域反散射问题的研究，主要研究内容包括：（1） 时域波动方程快速成像算法的数学理论与数值方法;（2）多频数据反演波源的Fourier展开方法; （3）无相位反散射问题的唯一性理论与重构算法; （4）频域点源反演的直接成像算法。上述研究内容不仅可以丰富反散射问题的数学理论，同时可以为发展新型数值计算方法提供有益的参考。