A Proposal for Research in Waves in Random Media and Applications

随机介质中的波研究及其应用的提案

• 批准号: 9971972
• 负责人: George Papanicolaou
• 金额: $ 29.4万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-15 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971972&HistoricalAwards=false
• 关键词: Proposal; Research; Waves; Random; Media

Papanicolaou9971972 The goal of this project is to develop and use analyticaland computational methods for geophysical wave propagation. Theinvestigator and his colleagues have developed two generaltheories: one based around radiative transport for elastic wavepropagation in the crustal waveguide, and the other based onwaves in nearly layered random media. The mathematical issuesinvolved in radiative transfer are (i) the derivation of boundaryconditions (ii) the introduction and use of good Monte Carlomethods (iii) the solution of realistic inverse problems forparameter identification. In the case of nearly layered media animportant mathematical issue is the robustness of wavelocalization, known to hold for perfectly layered random media.Another direction of study is the assessment of the effect ofrandom inhomogeneities on time reversal imaging, called wavemigration in geophysics. The mathematical analysis requires timedomain statistics for time reversed pulses, which have to beconstructed by Fourier analysis. Theoretical explanations areprovided for the increased resolution in time reversal imagingdue to the inhomogeneities. Geophysical wave propagation, the propagation of seismicwaves in the earth's crust for example, requires mathematicalmodeling at several levels depending on the accuracy and theresolution required. It is impossible to simulate numericallyseismic wave propagation over a few hundreds of kilometers of theearth's crust while taking into account the detailedinhomogeneities in the earth's crust. Radiative transport theorycan be used very effectively and at much lower computationalcost, and it can provide estimates for the codas (tails) ofseismograms with enough accuracy to interpret and analyze manycontinental seismic events. The project provides the mathematicalmethodology and infrastructure for using radiative transport ingeophysical problems. The investigator looks carefully at what isneeded to deal with discontinuities and interfaces in the crustalenvironment, what is needed for large scale imaging from surfaceseismic measurements, what is needed for assessing the waveguideeffects that appear in long distance seismic wave propagation,and what is needed for the effective numerical simulation of longdistance seismic wave propagation.

帕帕尼科拉9971972该项目的目标是开发和使用地球物理波传播的分析和计算方法。这位研究人员和他的同事们发展了两种普遍的理论：一种基于弹性波在地壳波导中传播的辐射传输，另一种基于近乎分层的随机介质中的波。辐射传递所涉及的数学问题是：(1)边界条件的推导；(2)良好蒙特卡罗方法的引入和使用；(3)参数识别的实际反问题的求解。对于接近分层的介质，一个重要的数学问题是小波变换的稳健性，众所周知，它适用于完全分层的随机介质。另一个研究方向是评估随机非均质性对时间反转成像的影响，在地球物理中称为波迁移。数学分析需要时间反转脉冲的时域统计，这必须通过傅里叶分析来构造。从理论上解释了时间反转成像中由于不均匀而提高分辨率的原因。地球物理波的传播，例如地震波在地壳中的传播，需要在几个层次上进行数学建模，这取决于所需的精度和解决方案。在考虑地壳中细微的不均质性的同时，不可能用数值方法模拟地震波在几百公里的地壳中的传播。辐射输运理论可以非常有效地使用，而且计算成本低得多，它可以提供对地震记录尾部(尾部)的估计，并具有足够的精度来解释和分析许多大陆地震事件。该项目为在地球物理问题中使用辐射传输提供了数学方法和基础设施。研究人员仔细研究了处理地壳环境中的不连续性和界面需要什么，从地表地震测量中进行大规模成像需要什么，评估长距离地震波传播中出现的波导效应需要什么，以及有效地对长距离地震波传播进行数值模拟需要什么。