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

该项目涉及到各种所谓的时频（尽管在这种情况下，更多的是空间定位/空间频率）技术在地震断层扫描中的应用，以便捕捉广泛、平滑的特征和定位良好的突变或尖状特征。在第一阶段，pi计划对问题的数学复杂性进行仔细的、逐步的研究。他们将构建适当的小波或类小波基，并利用目前对如何有效表征相对于这些基的稀疏数据或结构的新兴理解。他们期望，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复杂的、自适应的技术相当的结果，代价是问题“大小”的对数因素，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，然后是逆问题。在第二阶段，他们希望将其与Gilbert、Muthukrishnan、Strauss及其同事正在开发的超快算法结合起来，或者与Coifman、Laffont和Maggioni开发的图形扩散技术和相关的多分辨率结构结合起来。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，他们将把地震波形（而不是传播时间）反演的积分核发展成一个浓缩的小波基。这是分割三维波形层析问题的关键步骤。它应该允许他们将N个地震图的（难以管理的）大型非线性逆问题拆分为N个可管理的非线性优化问题。在地震层析成像中，地球物理学家通过对地震或爆炸激发的地震波的数学研究，并通过地震仪网络记录下来，试图获得有关地球隐藏特征的信息。这在某种程度上类似于我们更熟悉的计算机断层扫描，在这种扫描中，医生通过从多个角度拍摄透射的x射线照片来获取病人身体内部的信息。这种性质的问题被称为“逆问题”。另一个逆向问题是图像的去模糊；对于这个问题，小波变换，一种最近发展起来的数学工具，已经显示出特别好的适应性，当人们寻求“重建”的对象在区域之间有明显的边界时，否则是光滑的。地球内部的结构，如俯冲洋底或相变，同样可以有明显的边界，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换到地震层析成像是不可能的，因为地球物理反演问题通常规模更大、更复杂，其中的数据不能在一个漂亮的矩形网格上收集，而且台站覆盖通常存在很大的缺口。整个项目将结合前沿的应用数学技术和有限频率地震层析成像这一新兴领域的新发展。科学家们希望地球物理学和数学研究者之间的合作最终能获得更清晰、更准确的地球深部结构图像。

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.