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

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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.