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

研究的重点是描述波与随机介质在空间和时间上的相互作用。波的散射（远场相互作用）有许多重要的物理和工程应用，包括雷达、天文学和无线通信。散射总体的统计特性在描述散射过程中起着至关重要的作用。人口可能以不同的方式实现，这取决于问题——在海事雷达中是海洋表面的面，在天文学中是大气，在无线中是微波可用传播路径的数量。然而，每种情况都可以用大致相同的数学术语来描述。对于相互作用的种群，我们应用新的数学技术来更好地理解它们的动态和大时间行为。这项工作利用了理论物理学中的基本技术——伊辛模型和路径积分方法，与随机微积分相结合，来描述这些种群及其与各种波的相互作用。这些研究对雷达应用，特别是与地热成像和气候学相关的卫星干涉合成孔径雷达（InSAR）具有重要意义。在无线通信中，这些技术与非固定接收机-发射机环境中的时变信道具有重要的相关性，具有编码和信道容量计算的含义。声学传播和散斑现象的新应用包括声纳，用于生成合成图像（例如，与海啸预测相关的海底图像），以及通过随机推理技术进行超声医学成像。在核磁共振（近场相互作用）中，感兴趣的填充由自旋组成，其振幅为自旋噪声。化学研究应该在成像应用方面产生重要影响，特别是在功能磁共振成像方面，这对加拿大的健康研究界至关重要。总之，该研究解决了波在随机环境中的行为，并利用这种描述来提取各种物理应用中的有用信息。

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.