A Proposal for Research in Waves in Random Media and Applications

这个项目的目标是发展和使用地球物理波传播的分析和计算方法。研究者和他的同事们发展了两种一般理论：一种是基于地壳波导中弹性波传播的辐射输运，另一种是基于近分层随机介质中的波。辐射传递中涉及的数学问题是(i)边界条件的推导（ii）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。在近分层介质的情况下，一个重要的数学问题是小波化的鲁棒性，已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不均匀性对时间反转成像的影响，在地球物理学中称为波偏移。数学分析需要对时间反转脉冲进行时域统计，而时间反转脉冲必须通过傅里叶分析来构造。对非均匀性导致的时间反演成像分辨率的提高提供了理论解释。地球物理波的传播，例如地震波在地壳中的传播，需要根据精度和所需的分辨率在几个层次上进行数学建模。在考虑到地壳的详细均匀性的情况下，不可能在数值上模拟地震波在地壳几百公里内的传播。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的估计，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输运提供了数学方法和基础设施。研究者仔细研究了处理地壳环境中的不连续面和界面所需要的东西，地表地震测量的大规模成像所需要的东西，评估长距离地震波传播中出现的波导效应所需要的东西，以及长距离地震波传播的有效数值模拟所需要的东西。

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.