Mathematical analysis and inverse theory for seismic and medical imaging

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The objective of this program is to develop accurate mathematical theories and efficient algorithms for improving the resolution of seismic and medical images.***While the applications in seismic (earth) and medical (human) imaging are very different, the physical and mathematical principles behind the technologies are very similar. Energy is transmitted into the body under study (earth, or human) in the form of sound waves, or electromagnetic waves, and reflected off objects in the interior to recording devices on the surface. Complex mathematical algorithms are applied to the data and an image is created that can be viewed on the screen. Such images are used to identify oil and gas deposits in the earth, or the presence of organs, injuries, or even tumours within the human body.***Seismic imaging needs to be improved so that not just geological structures can be identified, but also their exact location and the nature of the rocks and fluids contained therein. This will improve the economics of oil and gas discovery and recovery, as well as advance the technology of carbon capture and storage in the earth's subsurface. ***Similarly, medical imaging needs to be improved so that higher resolution, dynamic images can be generated, to better identify and diagnose pathologies inside the body. By using non-invasive data collection such as ultrasound, or tissue sensing adaptive radar, medical teams can diagnose and prescribe treatment without harmful, invasive procedures on the human body.***The focus of this research is the development of  mathematical theories and computational methods that makes improvements in this imaging possible. ***We develop novel tools from functional and harmonic analysis, pseudodifferential operators, integral operators, and other spectral methods, to better model the physical processes of wave propagation, viscoelastic effects of the earth and of living tissue, as well as signal generation and capture, to improve the analysis and processing of the physical data gathered in industrial seismic surveys and practical medical instruments. Mathematical inverse theory provides a promising approach to tackle these problems in a uniform, advanced manner. ********