弹性波多频反源问题的Fourier方法研究

The inverse source problems have attracted considerable attention in scientific fields and engineering applications, such as antenna synthesis, medical imaging, remote sensing and seismic monitoring. This project is devoted to an inverse source problem for the elastic wave, that is, reconstructing an unknown elastic source from the multi-frequency measurements. Although many iterative algorithms have been developed for reconstructing the elastic source, fewer non-iterative reconstruction methods have been established in the frequency domain. In this project, we mainly consider a non-iterative reconstruction approach and give some theoretical analysis of the algorithm. First, based on the Fourier series expansion, we develop a novel fast and non-iterative reconstruction method for solving the problem. A key ingredient of this method is that it utilizes the decomposition of the elastic wave, and tranfers the coupled P wave and S wave into a Helmhotz system by choosing special frequencies and coefficients. Second, based on the Fourier analysis, theoretical feasibility and stability analysis are provided for the reconstruction method. Meanwhile, numerical experiments are presented to illustrate the effectiveness of the proposed method in three dimensions. We believe that the expected results of this project will not only enrich the studies on the inverse source problems of the elastic wave, but also serve a useful motivation for practical applications.

波源反演在实际工程中具有广泛应用，如：声呐定位、医学成像、地球遥感和地震监测。本项目研究一类弹性波多频反源问题，即利用边界测量的多频位移场信息重构未知的弹性波源项。虽然已有很多频域迭代算法用于弹性波源的反演，但相关的频域非迭代算法的研究还很不完善。本项目从非迭代反演方法的构造和重构算法的分析这两方面出发，主要研究内容包括：一、基于Fourier级数展开，设计一种非迭代快速反演方法。其核心是先将弹性波矢量分解成P波和S波，再通过选取特殊频率和系数将P波和S波重新耦合，使得耦合的波场满足Helmhotz方程。二、基于Fourier分析，着重阐明重构算法可行性的数学理论基础，分析算法的稳定性，并通过三维数值算例验证算法的有效性。本项目的研究成果不仅能够丰富弹性波反源问题的数值方法，而且对实际应用也具有一定的参考价值。

反源问题主要是根据边界测量的波场信息反演波源的位置和强度等物理参数，相关研究不仅具有重要的科学意义，而且具有潜在的实用价值。本项目主要围绕弹性波反源问题，分别开展了L2源和矩张量点源的反演重构，研究内容包括：（1）多频数据反演L2源的Fourier方法；（2）多频数据反演矩张量点源的直接采样方法；（3）波源反演在隐形和超分辨成像中的应用。本项目取得了一系列重要的研究成果：一、基于解耦和耦合的策略，提出了反演弹性波L2源的Fourier方法，创造性地实现了三维弹性波源的快速高精度重构。二、基于贝塞尔函数渐近展开的特性，构造了反演矩张量点源的直接采样方法，首次给出了多频叠加的收敛性分析。三、将多频反源思想应用于反散射问题，在探测未知散射体中，不但能够从理论上进行隐形，而且可以实现超分辨率反演重构。基于上述研究工作，本人依托本项目共发表SCI论文7篇。特别地，本项目的研究成果不仅能够丰富反源问题的数值方法，而且对实际应用也具有一定的参考价值。