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

该项目涉及在地震层析成像中应用各种所谓的时频(尽管在本文中是空间局部化/空间频率)技术，以捕捉广泛、平滑的特征和精确定位的突变或尖峰特征。在第一阶段，PI计划对这个问题的数学复杂性进行仔细、循序渐进的研究。他们将构造适当的小波或类小波基，并利用目前正在出现的关于如何有效地表征相对于这些基稀疏的数据或结构的理解。他们预计，最近的数学进展表明，简单的、非适应性的方法可以给出与更花哨的适应性技术相媲美的结果，代价是问题的“大小”是对数的，这将帮助他们在项目的下一阶段建立一个好的方法，首先攻击正问题，然后攻击反问题。在第二阶段，他们希望将其与Gilbert、Muthukrishnan、Strauss和他的同事正在开发的超快算法相结合，和/或与由Coifman、Laffont和Maggioni开发的图形扩散技术和相关的多分辨率结构相结合。随着新技术的发展，这些新技术将应用于正在进行的全球地震层析成像研究项目。最后，他们将把用于地震波形(而不是旅行时)反演的积分核发展成一个压缩小波基。这是解决波形层析三维问题的关键一步。它应该允许他们将N个地震记录的(无法管理的)大而非线性的反问题分成N个规模可管理的非线性优化问题。在地震层析成像中，地球物理学家试图通过对地震或爆炸激发的地震波进行数学研究，并由地震仪网络记录下来，获得有关地球隐藏特征的信息。这在某种程度上类似于更常见的CAT扫描断层扫描，即医生通过从多个角度拍摄传输的X射线照片来寻找关于患者体内的信息。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，最近发展起来的一种数学工具--小波变换--被证明特别适用于当一个人试图“重建”的对象在原本是平滑的区域之间具有清晰的边界时。地球内部的构造，如俯冲洋底或相变，同样可以有尖锐的边界，而物理性质的其他变化可能分布得更平稳。因此，小波技术在地震层析成像中也可能特别有用。然而，仅仅将图像去模糊技术转置到地震层析成像是不可能的，因为地球物理反问题的规模和复杂性通常要大得多，其中的数据不能在漂亮的矩形网格上收集，并且通常存在着台站覆盖的大间隙。整个项目将把尖端的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对地球深层结构的更清晰和更准确的图像。

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.