Mathematical modelling and computational methods for imaging and advanced sensor technology

我们的研究创建了数学模型、方法和算法，以支持地震、医学和工业成像传感技术的进步。尤其令人感兴趣的是新的传感器技术和计算能力带来的挑战，这些技术允许在数学分析和数值方法中使用更好、更深入的技术。 大多数人都熟悉成像技术，如医用超声波和CAT扫描，以查看人体内部，以及机场安检处的X光机，可以“看到”我们的行李。相关技术被用于工业，如用于确定油层位置的地球地下地震成像，或用于非破坏性材料测试，如用于穿透墙壁以找到光束、管道和电线的基于微波的“螺柱探测仪”。 人们不太熟悉这样一个事实，即这些成像技术需要高性能的数学和计算方法来转换从传感器(超声波、X射线、地震)收集的数据，并将它们转换为被调查物理对象(人体、机场行李、地球地下)的有用图像。 我们的研究是发展支持这些成像方法所需的数学。这包括创建所研究物理系统的准确数学模型、记录物理信号的传感器模型、将信号传播描述为数学线性运算符的函数分析、表征和分解这些信号的谐波分析，以及支持快速、准确计算的数值方法。用于随机放置传感器的压缩传感；用于自动检测和定位微地震压裂、入侵者和交通模式等事件的计算神经网络。最优传输理论用于支持成像中的数学反问题。 其中一个焦点是新型分布式声波传感器(DAS)，这种传感器将光纤电缆埋在地下，以监控安全围栏、铁路线或油井现场。虽然光纤是一种光学设备，但物理振动会拉伸光纤，然后通过激光干涉仪的精确测量进行检测。这就把光纤变成了声学传感器。DAS值得注意的是，一条廉价的光纤电缆可以沿着数十公里的基础设施安装，并可以用来对这几十公里沿线的事件进行成像。DAS有效地创建了一个可用于成像算法和事件监控的连续“虚拟地震检波器”流。 应用包括提高成像质量，增强对图像的地球物理或医学解释，并允许在新的情况下引入新的传感器技术，如对石油生产地点、管道和二氧化碳储存设施的DAS监测。高素质人才的培训在该计划中发挥着核心作用。 这项研究利用分析和数值计算的最佳结果，支持传感器和成像领域的不断进步的技术。

Our research creates mathematical models, methods and algorithms to support advances in sensing technologies for imaging -- seismic, medical and industrial. Of particular interest are challenges arising with new sensors technology and computational power that permit better, deeper techniques in mathematical analysis and numerical methods. Most people are familiar with imaging technologies such as medical ultrasounds and CAT scans to see inside the human body, as well as X--ray machines at airport security that can “see” inside our luggage. Related technologies are used in industry, such seismic imaging of the earth's subsurface for locating oil reservoirs, or non-destructive materials testing like microwave-based "stud finders" used to see through walls to find beams, pipes, and wires. People are less familiar with the fact that these imaging technologies require high powered mathematical and computational methods to transform data collected from sensors (ultrasonic, X--ray, seismic) and turn them into useful images of the physical object under investigation (human body, airport luggage, earth's subsurface). Our research is on developing the mathematics necessary to support these imaging methods. This includes creating accurate mathematical models of the physical systems under study, models of the sensors recording physical signals, functional analysis to describe signal propagation as mathematical linear operators, harmonic analysis to characterize and decompose these signals, and numerical methods to support fast, accurate computation. Compressive sensing used for randomized placement of sensors; computational neural nets to automatically detect and localize events such as microseismic fracturing, intruders, and traffic patterns. Optimal transport theory is used to support mathematical inverse problems in imaging. One focus are novel distributed acoustic sensors (DAS), where a fibre optic cable is buried in the ground to monitor a security fence, rail line or oilwell site. While the fibre is an optical device, physical vibrations stretch the fibre, then detected with precise measurements of a laser interferometer. This turns the optical fibre into an acoustic sensor. DAS is remarkable as a cheap fibre cable can be installed along tens of kilometres of infrastructure and can be used to image events all along these many kilometres. DAS effectively creates a continuous stream of “virtual geophones” that can used in imaging algorithms and event monitoring. Applications include improving the quality of imaging, to enhance geophysical or medical interpretation of the images, and allow introduction of new sensor technologies in novel situations such as DAS monitoring of oil production sites, pipelines, and CO2 storage facilities. Training of highly qualified personnel plays a central role in the program. This research supports ever advancing technologies in sensor and imaging, using the best results from analysis and numerics.