Mathematical analysis and inverse theory for seismic and medical imaging

该计划的目标是发展准确的数学理论和有效的算法，以提高地震和医学图像的分辨率。虽然地震（地球）成像和医学（人体）成像的应用非常不同，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递到被研究的身体（地球或人类），并从内部物体反射到表面的记录设备。将复杂的数学算法应用于数据，并创建可以在屏幕上查看的图像。这些图像被用来识别地球上的石油和天然气矿床，或人体器官、损伤甚至肿瘤的存在。***地震成像需要改进，这样不仅可以识别地质构造，而且可以识别它们的确切位置和其中所含岩石和流体的性质。这将提高石油和天然气发现和开采的经济效益，并推进地球地下碳捕获和储存技术。***同样，医学成像也需要改进，以便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，如超声波或组织感应自适应雷达，医疗团队可以诊断和开出治疗方案，而无需对人体进行有害的侵入性手术。这项研究的重点是数学理论和计算方法的发展，使这种成像的改进成为可能。***我们开发了新的工具，从泛函和谐波分析，伪微分算子，积分算子，和其他频谱方法，以更好地模拟波的传播，地球和活组织的粘弹性效应的物理过程，以及信号的产生和捕获，以提高在工业地震调查和实际医疗仪器收集的物理数据的分析和处理。数学逆理论为解决这些问题提供了一种统一的、先进的方法。* * * * * * * *

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. ********