Hyperbolic Inverse Problems in Random Environments

这个项目的目标是获得复杂（杂乱）环境中波成像的严格数学理论。该主题位于数学与物理、概率、数值模拟和信号处理的界面。该研究是由应用领域的挑战驱动的，如探地雷达、卫星成像和大气湍流跟踪、老化混凝土等材料的无损超声检测、浅水成像和地下勘探。复杂介质在此类应用中无处不在，对成像过程造成了严重的阻碍，而这在很大程度上被现有成像技术所忽视。此外，复杂介质中波浪传播的计算机模拟面临着巨大的计算挑战。为了进一步发展成像技术，需要对这种介质复杂的散射效应进行数学分析。该项目旨在分析声波和电磁波在复杂介质中的远程传播，这些介质也可能随时间变化，开发新的自适应成像方法来减轻介质散射效应，并提出可以改善成像过程的新测量装置。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。从这些随机过程的不确定性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更好地理解声波方程和麦克斯韦方程组的这个问题。用渐近随机分析和不变嵌入相结合的数学方法来研究传输算子和反射算子。该项目考虑混合静态或随时间变化的随机过程。时间变化在各种设置中进行了研究，时间变化与探测复杂环境的传感器发出的脉冲持续时间有关。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。该方法应能够检测到被测波在环境中散射导致的相干性损失，并通过过滤在成像中无用的场的成分来提高信噪比。对成像方法的分析寻求一种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.