This award supports a research program aimed at designing mathematically-informed computational tools for processing large, high-dimensional seismic datasets that display directional structure along lower-dimensional manifolds. The progress that occurred over the past few decades in seismic imaging has largely ignored growing data-related complications, such as coherent noise, multiple scattering, irregular acquisition geometries, and simultaneous acquisition. Computational harmonic analysis provides solutions to these problems by formulating optimization problems that leverage sparsity in a transformed domain. These tools can however not be relied upon for very large scale inversion tasks, because they are not computationally advantageous in such regimes. This project revisits the mathematical underpinnings of multiscale directional transforms with a view toward designing low-redundancy, high-dimensional architectures that should be competitive for even the most data-intensive inversion scenarios.
Moore's law of exponential increase in computing performance is not often matched by exponential progress in the computational sciences. The culprit is the lack of scalability of mainstream algorithms: the size of problems that can be solved grows more slowly than hardware capabilities. In increasingly many applications, the input of mathematicians is needed to help engineers and applied scientists rethink the design of numerical codes to avoid this curse of scalability. This project is an effort to take a step back and introduce new algorithmic ideas for seismic imaging, the discipline concerned with imaging the subsurface of the Earth. Seismic imaging is the energy sector's main predictive tool for hydrocarbon, water, and geothermal energy prospection. It is at the heart of monitoring techniques for reservoirs and carbon sequestration experiments. It has proved useful to geophysicists who debate the geological composition of the Earth's mantle. High-resolution seismic imaging is also starting to enable the Army and the Air Force to detect IEDs. All these remote imaging problems have by now become formidably complex computational questions that our generation will be responsible for solving.
We have worked on developging fast algorithms for computing general multilinear operators. Multilinear operators have been intensively studied in harmonic analysis and have important applications in computational geosciences. However, there has been little research on their numerical application. In this project, we have developed the first general linear complexity algorithm for applying multilinear multiplier operators. The algorithm combines the ideas and techniques from hierachical domain decomposition, low rank approximation, and Fourier transform. The algorithm has been applied to some prototype problems from computational geophysics. We have also worked on adaptive analysis of higher dimensional data. Emphrical mode decomposition (EMD) has been proven to be a powerful tool for decomposing complex signals into intrinsic mode functions. One recently developed tool for EMD is the synchrosqueezing wavelet transform for 1D signal analysis proposed by Daubechies et al. In this project, we developed the synchrosqueezed wave atom and synchrosqueezed curvelet transforms, which are a higher dimensional generalization that can potentially have important applications in seismic data analysis. These new transforms combine the synchrosqueezing technique with the PI's previous work on wave atoms and curvelets. The resulting transforms exhibit much better capability in decomposing superposed wave form signals into individual components. These transforms have been implemented faithfully and have been applied to problems from seismic signal processing
