CMG: When Sparse Meets Dense: New Mathematical Approximations Applied to Seismic Tomography

CMG：当稀疏遇到密集：应用于地震层析成像的新数学近似

• 批准号: 0530865
• 负责人: Ingrid Daubechies
• 金额: --
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-15 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0530865&HistoricalAwards=false
• 关键词: CMG; When; Sparse; Meets; Dense

The project concerns the application of various so-called time-frequency (although in this context rather spatial localization/spatial frequency) techniques to seismic tomography, in order to capture both broad, smooth features and well-localized abrupt transitions or spiky features. In a first stage, the PIs plan to make a careful, stepwise study of the mathematical intricacies of the problem. They will construct appropriate wavelets or wavelet-like bases, and take advantage of the presently emerging understanding of how to characterize effectively data or structures that are sparse with respect to these bases. They expect that recent mathematical progress that shows simple, non-adaptive methods can give results comparable to fancier, adaptive techniques, at the cost of a logarithmic factor in the "size" of the problem, will help them in building a good approach in the next stage of the project, to attack first the direct, and then the inverse problem. In a second stage they hope to couple this with the ultrafast algorithms that are being developed by Gilbert, Muthukrishnan, Strauss and co-workers, and/or with the graph-diffusion technique and the associated multiresolution structure developed by Coifman, Laffont and Maggioni. The new techniques will be applied in ongoing global seismic tomography research projects as they are developed. Finally, they will develop the integral kernels for the inversion of seismic waveforms (as opposed to travel times) into a condensed wavelet basis. This is a crucial step in the partitioning of the three-dimensional problem of waveform tomography. It should allow them to split the (unmanageably) large and nonlinear inverse problem for N seismograms into N nonlinear optimization problems of manageable size. In seismic tomography, geophysicists seek to obtain information about hidden features of the Earth from a mathematical study of seismic waves excited by earthquakes or explosions, and recorded by a network of seismographs. This is to some extent similar to the more familiar CAT-scan tomography, in which doctors seek information on the inside of the body of a patient by taking transmitted X-ray pictures from many angles. Problems of this nature are called "inverse problems". Another inverse problem is the deblurring of images; for this problem, the wavelet transform, a recently developed mathematical tool, has shown to be particularly well adapted when the object one seeks to "reconstruct" can have sharp boundaries between regions that are otherwise smooth. Structures inside the Earth, such as subducting ocean floors or phase transitions, can likewise have sharp boundaries, whereas other variations in the physical properties may be distributed more smoothly. Wavelet techniques may therefore be particularly useful in seismic tomography as well. However, it is impossible to just transpose the image deblurring techniques to seismic tomography, because of the typically much larger size and greater complexity of the geophysical inverse problems, in which data can not be collected on a nice rectangular grid and large gaps in station coverage usually exist. The whole project will combine cutting edge applied mathematical techniques with new developments in the young field of finite frequency seismic tomography. The PIs expect that this collaboration between geophysical and mathematical investigators will eventually lead to sharper and more accurate images of the Earth's deep structure.

该项目涉及将各种所谓的时间-频率（虽然在这方面是空间定位/空间频率）技术应用于地震层析成像，以便捕捉广泛、平滑的特征和定位良好的突变或尖峰特征。在第一阶段，PI计划对问题的数学复杂性进行仔细的逐步研究。他们将构建适当的小波或小波基，并利用目前新兴的理解，如何有效地表征数据或结构，稀疏相对于这些基地。他们希望，最近的数学进展表明，简单的，非自适应的方法可以得到与更花哨的结果，自适应技术，在成本的对数因子的“大小”的问题，将有助于他们在建立一个良好的方法在下一阶段的项目，攻击第一个直接，然后逆问题。在第二阶段，他们希望将其与吉尔伯特、穆图克里希南、施特劳斯及其同事正在开发的超快算法相结合，或者与图扩散技术和Coifman、Laffont和Maggioni开发的相关多分辨率结构相结合。这些新技术将在开发完成后应用于正在进行的全球地震层析成像研究项目。 最后，他们将开发用于地震波形反演的积分核（与旅行时间相反）到压缩小波基中。这是波形层析成像的三维问题的划分中的关键步骤。它应该允许他们将N个地震图的（无法管理的）大型非线性逆问题分解为N个可管理大小的非线性优化问题。在地震层析成像中，地震学家试图从地震或爆炸激发的地震波的数学研究中获得有关地球隐藏特征的信息，并由地震仪网络记录。这在某种程度上类似于更熟悉的CAT扫描断层扫描，其中医生通过从多个角度拍摄透射X射线照片来寻找患者体内的信息。这种性质的问题被称为“逆问题”。另一个逆问题是图像的去模糊;对于这个问题，小波变换，最近开发的数学工具，已被证明是特别好地适应时，一个人试图“重建”的对象可以有尖锐的边界之间的区域，否则是光滑的。 地球内部的结构，如俯冲海底或相变，同样可以有尖锐的边界，而其他物理性质的变化可能分布得更平滑。因此，小波技术在地震层析成像中也可能特别有用。然而，这是不可能的，只是转的图像去模糊技术，地震层析成像，因为通常更大的规模和更大的复杂性的地球物理反演问题，其中的数据不能收集在一个很好的矩形网格和大的差距，在站覆盖通常存在。 整个项目将把联合收割机尖端应用数学技术与有限频率地震层析成像这一年轻领域的新发展结合起来。PI预计，地球物理和数学研究人员之间的这种合作最终将导致地球深层结构的更清晰，更准确的图像。