Proposal on Pulse Shaping for Waves in Multiscale Media

索尔纳0093992 研究者分析了传播波脉冲在介质中的粗糙变化。 从介质的“粗糙度”信息中预测波脉冲的演化是一个具有深刻数学意义和实际意义的问题，从透射脉冲的测量中推断介质信息的逆问题也是如此。 大多数情况下，对介质的（精细尺度）变化的详细描述是不可能的，只能描述它们的统计特征。 因此，研究者将细尺度介质变化看作是随机过程的一种实现。 这就需要分析具有随机变化系数的微分方程。 该项目有三个具体目标：第一，获得一个表征的变换声脉冲传播的介质中的一个连续的长度尺度，一个媒体定义的随机分形。 其次，将波脉冲变换理论推广应用到一些具体的反演问题中，特别是高对比度地震成像和时间反转镜。 第三，在不知道这些变化细节的情况下，推导出电磁脉冲如何受到介质非均匀性影响的严格界限。 研究者要解决的主要问题是：当变化不小时，传播波在变化的介质中是如何表现的？ 例如，当一束激光在湍流大气中传播时，它是如何受到强烈的、湍流的、局部传播速度变化的影响的？当这样的介质变化是平滑的，比如传播速度恒定时，问题就很容易理解了。 对于粗糙介质的变化，如上面的例子，波脉冲与介质不均匀性的相互作用是一个基本的，但在很大程度上是开放的问题。 预测信号或波脉冲在粗糙介质中的演化是一个具有深刻数学意义和实际意义的问题。 除了湍流大气中的波传播外，还有许多其他的应用. 在反射地震学中，用波脉冲探测地壳，反射的脉冲被用来推断介质中宏观尺度变化的信息。 因此，了解介质中粗糙的微尺度变化如何影响探测脉冲是很重要的。 遥感、海洋声学和医学成像提供了类似的例子。 在通信系统中，能够描述介质的缺陷和变化如何影响传播信号是很重要的。 在复合材料理论中，重要的是优化构造复杂材料，使其具有某些期望的传播特性。

Solna0093992 The investigator analyzes how a propagating wave pulsebehave in a medium with rough variations. Predicting theevolution of the wave pulse from information about the`roughness' of the medium is a problem of deep mathematical andpractical interest, as is the inverse problem of deducinginformation about the medium from measurements of a transmittedpulse. Most often a detailed description of the (fine scale)variations of the medium is not possible, only a description oftheir statistical character. Hence, the investigator considersthe fine scale medium variations as a realization of a stochasticprocess. This entails analysis of differential equations withrandomly varying coefficients. The project has three specificobjectives: First, obtain a characterization of thetransformation of sound pulses propagating in a medium with acontinuum of length scales, a medium defined in terms of a randomfractal. Second, extend and use theory about the transformationof a wave pulse in some specific inverse problems, in particular,high contrast seismic imaging and time reversal mirrors. Third,derive strict bounds for how an electromagnetic pulse can beaffected by the medium heterogeneity without knowing the detailsof these variations. The main question the investigator addresses is: How does apropagating wave behave in a medium that varies when thevariations are not small? For instance, when a laser beampropagates through the turbulent atmosphere how is it affected byintense, turbulent, variations in the local speed of propagation?When such medium variations are smooth, say with a constantspeed of propagation, the problem is well understood. For roughmedium variations, as in the above example, the interaction of awave pulse with the medium heterogeneities is a fundamental yetlargely open question. This problem, predicting the evolution ofsignal or wave pulse that propagates in a rough medium, is ofdeep mathematical and practical interest. In addition to wavepropagation in the turbulent atmosphere there are many otherapplications. In reflection seismology the earth's crust isprobed with a wave pulse and the reflected pulses are used toinfer knowledge about macroscale variations in the medium. It isthen important to know how the rough microscale variations in themedium affect the probing pulse. Remote sensing, ocean acoustics,and medical imaging provide similar examples. In communicationsystems, it is important to be able to describe how mediumimperfections and variations affect the propagating signal. Inthe theory of composite materials, it is important to optimallyconstruct a complex material so that it has some desiredpropagation properties.