Proposal on Pulse Shaping for Waves in Multiscale Media

索尔纳0093992研究人员分析了传播波脉冲在具有粗略变化的介质中的行为。根据介质的“粗糙度”信息预测波脉冲的演化是一个非常有数学意义和实际意义的问题，同样的问题也是从测量传输的脉冲中推断出介质信息的逆问题。大多数情况下，无法详细描述介质的(细微尺度)变化，只能描述其统计特性。因此，研究者认为细尺度介质变化是随机过程的一种实现。这就需要对随机变系数的微分方程进行分析。该项目有三个具体目标：第一，获得声脉冲在具有连续长度尺度的介质中传播的变化的特征，该介质是根据随机分形定义的介质。其次，将波脉冲变换理论推广和应用于一些具体的反问题，特别是高对比度地震成像和时间反转镜。第三，在不知道这些变化的细节的情况下，推导出电磁脉冲如何受到介质非均质性影响的严格界限。研究人员解决的主要问题是：当变化不小时，传播波在变化的介质中表现如何？例如，当一束激光在湍流大气中传播时，它是如何受到局部传播速度剧烈的、湍流的变化的影响的？当这样的介质变化是平滑的时，比如说以恒定的传播速度，这个问题就很好理解了。对于粗糙介质的变化，如上面的例子，a波脉冲与介质非均质性的相互作用是一个基本的尚未解决的问题。预测信号或波脉冲在粗糙介质中传播的演化是一个具有深远的数学意义和实用价值的问题。除了波在湍流大气中的传播外，还有许多其他的应用。在反射地震学中，用波脉冲探测地壳，反射脉冲用来推断介质中宏观尺度的变化。因此，重要的是要知道介质中粗略的微尺度变化如何影响探测脉冲。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，最重要的是优化构造一种复合材料，使其具有期望的传播性能。

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.