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整个空间和时间中的相互作用。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。散射粒子群的统计特性在描述散射过程中起着至关重要的作用。根据问题的不同，人口可以以不同的方式实现-在海事雷达中作为海洋表面的小平面，在天文学中作为大气层，在无线电中作为微波可用传播路径的数量。然而，每种情况都可以用几乎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。这项工作借鉴了理论物理学中的基本技术-伊辛模型和路径积分法，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有关的卫星干涉合成孔径雷达（干涉合成孔径雷达）具有重要意义。在无线通信中，这些技术与非静止接收器-发射器环境中的时变信道具有重要的相关性，其涉及编码和信道容量计算。在声传播和斑点现象中的新应用包括声纳，通过随机推理技术生成合成图像（例如与海啸预测相关的海底图像）和超声医学成像。在NMR（近场相互作用）中，感兴趣的布居由自旋组成，并且所产生的振幅是自旋噪声。化学研究应在成像应用中产生重要影响，特别是在功能性磁共振成像中，对加拿大卫生研究界具有重要意义。总之，该研究解决了随机环境中波的行为，并利用这种描述在各种物理应用中提取有用的信息。

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.