Hyperbolic Inverse Problems in Random Environments

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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.