Stochastic differential equation and random field theory applied to waves in random media

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The focus of the research is the description of waves interacting with random media, throughout space and time. Scattering of waves (far field interaction) has many important physics and engineering applications, including radar, astronomy and wireless communications. The statistical characteristics of a scattering population play a crucial role in descriptions of scattering processes. The population may be realized in different ways depending on the problem - in maritime radar as facets on the ocean surface, in astronomy as the atmosphere, and in wireless as the number of available propagation paths for the microwave. Each case can be described in much the same mathematical terms, however. For populations that interact with each other we apply new mathematical techniques to better understand their dynamics and large time behaviour. The work draws on fundamental techniques in theoretical physics - the Ising model and the path integral method, in tandem with the stochastic calculus, to describe such populations and their interaction with waves of a variety of kinds. These studies are of vital significance in radar applications, especially satellite interferometric synthetic aperture radar (InSAR) with relevance to geothermal imaging and climatology. In wireless communications the techniques have important relevance to time varying channels in non-stationary receiver-transmitter environments with implications for coding and channel capacity calculations. Novel applications in acoustic propagation and speckle phenomena include sonar, in the generation of synthetic images (e.g. of the ocean floor with relevance to tsunami prediction), and ultrasound medical imaging, via stochastic inference techniques. In NMR (near field interaction), the population of interest consists of spins, and the resulting amplitude is spin noise. Chemical studies should have important consequences in imaging applications, particularly in functional magnetic resonance imaging, of key benefit to the Canadian health research community. In summary, the research addresses the behaviour of waves in random environments, and harnesses such description for extraction of useful information in a variety of physical applications.