A Proposal for Research in Waves in Random Media and Applications

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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.