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.
The problem of imaging the Earth's interior is currently one of society's most formidable technical challenges. It took geophysicists decades to be able to find good formulas to leverage the knowledge of seismic waves -- noise from earthquakes -- to create plausible maps of the subsurface. Yet, the quest for a solution of the so-called "inverse scattering" problem is not over: there exist situations in which no known imaging method would succeed, even when sufficient information is in principle present in the recorded seismic data. The problem is of a combinatorial nature, i.e., it owes to our computers' inability to perform exhaustive searches in high-dimensional spaces. The group of Prof. Demanet is focused on approaching this challenge from a mathematical angle, using techniques of modern optimization such as convex relaxation. With the help of this NSF grant, one postdoctoral scholarin the group were given the opportunity to investigate new, important computational solutions to the inverse problems of imaging. The new imaging methods developed in Prof. Demanet's group have scientific and industrial applications in 1) determining the structure of the Earth from seismic waves, 2) visualizing the interior of biological tissues with ultrasound waves, and 3) locating reflecting objects from microwaves in radar systems. The interdisciplinary topic of imaging (at the intersection of math, statistics, physics, and EE), though in high demand in industry, is not normally taught in graduate school. An important component of the grant's activity was the organization of two graduate summer schools to help train groups of young computational engineers in the common mathematical language of wave-based imaging.
