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.

Papanicolaou9971972 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。 研究人员和他的同事开发了两种通用理论：一种基于地壳波导中弹性波传播的辐射传输，另一种基于近层状随机介质中的波。 辐射传输涉及的数学问题是（i）边界条件的推导（ii）良好的蒙特卡罗方法的引入和使用（iii）参数识别的实际逆问题的解决。 在近层状介质的情况下，一个重要的数学问题是波定位的鲁棒性，已知这适用于完美层状的随机介质。另一个研究方向是评估随机不均匀性对时间反转成像的影响，在地球物理学中称为波迁移。 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建。 为时间反转成像中由于不均匀性而提高分辨率提供了理论解释。 地球物理波传播，例如地震波在地壳中的传播，需要根据所需的精度和分辨率进行多个级别的数学建模。 考虑到地壳中详细的不均匀性，不可能对地壳数百公里范围内的地震波传播进行数值模拟。 辐射传输理论可以非常有效地使用，并且计算成本要低得多，并且它可以以足够的精度对地震图的尾部进行估计，以解释和分析许多大陆地震事件。 该项目提供了利用辐射传输地球物理问题的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所需的内容、表面地震测量大规模成像所需的内容、评估长距离地震波传播中出现的波导效应所需的内容以及长距离地震波传播的有效数值模拟所需的内容。