Hyperbolic Inverse Problems in Random Environments

这个项目的目标是获得一个严格的数学理论成像波在复杂（杂乱）的环境。该主题位于数学与物理，概率，数值模拟和信号处理的接口。该研究是由应用领域的挑战驱动的，如探地雷达，通过大气湍流的卫星成像和跟踪，材料的无损超声检测，如老化混凝土，浅水成像和地下勘探。复杂介质在这些应用中无处不在，并且对成像过程构成严重障碍，这在很大程度上被当前成像技术所忽视。此外，波在复杂介质中传播的计算机建模面临着巨大的计算挑战。需要数学分析来揭示这种介质的复杂散射效应，从而可以改进现有的成像技术。该项目旨在分析声波和电磁波在复杂介质中的长距离传播，这些介质也可能随时间而变化，开发新的自适应成像方法，以减轻介质散射效应，并提出新的测量设置，以改善成像过程。复杂的环境自然是用随机过程建模的，波动方程具有随机系数和边界。不确定性从这些随机过程传播到成像应用中测量的波的不确定性是一个非常重要的问题。该项目的一个目标是开发一个更好地理解这个问题的声波方程和麦克斯韦方程组。数学是渐近随机分析和不变嵌入研究传输和反射算子的组合。该项目考虑混合静态或可能随时间变化的随机过程。在各种设置中研究了时间变化，相对于探测复杂环境的传感器发出的脉冲的持续时间，时间变化既快又慢。该项目的另一个目标是在复杂环境中开发一种新的鲁棒和自适应成像方法。该方法应该能够检测由于环境中的散射而导致的测量波的相干性的损失，并且通过过滤在成像中无用的场的分量来提高信噪比。成像方法的分析寻求量化图像聚焦及其统计稳定性的分辨率理论。

The objective of this project is to obtain a rigorous mathematical theory of imaging with waves in complex (cluttered) environments. The topic lies at the interface where mathematics meets physics, probability, numerical simulations, and signal processing. The research is driven by challenges in application areas such as ground penetrating radar, satellite imaging and tracking through atmospheric turbulence, nondestructive ultrasonic testing of materials such as aging concrete, imaging in shallow water, and underground exploration. Complex media are ubiquitous in such applications and pose a serious impediment to the imaging process, which is largely ignored by the present imaging technology. Moreover, computer modeling of wave propagation in complex media is faced with formidable computational challenges. Mathematical analysis is needed to unravel the complicated scattering effects of such media so that the present imaging technology can be advanced. This project seeks to analyze long-range propagation of sound and electromagnetic waves in complex media that may also vary in time, develop novel adaptive imaging methodologies that mitigate the medium scattering effects, and propose new measurement setups that can improve the imaging process. Complex environments are naturally modeled with random processes, and the wave equation has random coefficients and boundaries. The propagation of uncertainty from these random processes to the uncertainty of the waves measured in imaging applications is a highly non-trivial problem. One goal of the project is to develop a better understanding of this problem for the acoustic wave equation and Maxwell's system of equations. The mathematics is a combination of asymptotic stochastic analysis and invariant imbedding for studying transmission and reflection operators. The project considers mixing random processes that are static or may vary in time. The time variations are studied in various setups, with both rapid and slow time changes with respect to the duration of pulses emitted by sensors that probe the complex environment. Another goal of the project is to develop a novel robust and adaptive imaging methodology in complex environments. The methods should be able to detect the loss of coherence of the measured waves due to scattering in the environment and to enhance the signal to noise ratio by filtering the components of the fields that are useless in imaging. The analysis of the imaging methods seeks a resolution theory that quantifies the focusing of images and their statistical stability.