Mathematical analysis and inverse theory for seismic and medical imaging

该计划的目标是开发精确的数学理论和有效的算法来提高地震和医学图像的分辨率。*虽然在地震(地球)和医学(人体)成像中的应用截然不同，但这些技术背后的物理和数学原理非常相似。能量以声波或电磁波的形式传输到被研究的身体(地球或人类)，并从内部的物体反射到表面的记录设备。复杂的数学算法被应用于数据，并创建了可以在屏幕上查看的图像。这样的图像被用来识别地球上的石油和天然气矿藏，或者人体内器官、损伤甚至肿瘤的存在。*地震成像需要改进，以便不仅可以识别地质结构，而且可以识别它们的确切位置和其中包含的岩石和流体的性质。这将提高石油和天然气发现和回收的经济性，并推动地球地下碳捕获和储存技术的发展。*同样，医学成像需要改进，以便能够生成更高分辨率的动态图像，以便更好地识别和诊断体内的病理。通过使用非侵入性数据收集，如超声波或组织传感自适应雷达，医疗队可以在不对人体进行有害的侵入性程序的情况下诊断和开出治疗处方。*这项研究的重点是发展新的数学理论和计算方法，使这种成像的改进成为可能。*我们从泛函和调和分析、伪微分算子、积分算子和其他频谱方法中开发出新的工具，以更好地模拟波传播的物理过程、地球和活组织的粘弹性效应以及信号的生成和捕获，以改进对工业地震测量和实际医疗仪器中收集的物理数据的分析和处理。数学逆理论为以统一、先进的方式解决这些问题提供了一种很有前途的方法。********

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. ********