The investigator, with his students and collaborators, develops novel and efficient numerical methods for modeling high-frequency wave propagation and solution of associated inverse problems. These problems arise from seismic wave propagation, geometrical optics, optimal control, computerized tomography (CT), medical imaging, and material sciences. The proposal focuses on advancing fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods, a novel approach for this challenging problem, from theoretical, algorithmic, and practical perspectives. This interdisciplinary research complements the PI's educational goals by integrating education and research activities at undergraduate and graduate levels. Problems under consideration include exploring the deep connection between multiscale Gaussian beams and fast multiscale Gaussian wavepacket transforms to devise new algorithms for decomposing given data into Gaussian beams, developing new multiscale Gaussian beam methods for modeling acoustic, elastic and anisotropic waves, analyzing linearized inverse problems for acoustic and elastic wave equations by using multiscale Gaussian beams in the high frequency regime, devising novel algorithms to implement the resulting linearized inversion formulas, and validating the resulting algorithms by using synthetic data.
Modeling of high frequency waves is of great strategic value in diverse science and engineering disciplines, ranging from the US petroleum industry, seismic imaging, radar, sonar, medical imaging, remote sensing, submarine detection, material sciences to nanotechnology. The current surge in price for crude oil and other earth resources increasingly demands better imaging techniques in exploration seismology. The increasing amount of data in global and exploration seismology requires more sophisticated mathematical models. The techniques developed as part of this project provide crucial tools for the development of the next-generation seismic imaging tools with the potential to enable substantial cost savings in seismic explorations, expedite routine data processing, and protect the environment by optimizing drilling sites. Students from the PI's institution are involved in this innovative interdisciplinary research project.
This proposal focuses on advancing fast multiscale Gaussian wavepacket transforms and multiscale Gaussian beam methods for high frequency waves and inverse problems from theoretical, algorithmic and practical perspectives. The proposed interdisciplinary research complements the PIâ€™s educational goals by integrating education and research activities at undergraduate and graduate levels. Research objectives were to (1) explore the deep connection between multiscale Gaussian beams andfast multiscale Gaussian wavepacket transforms to devise new algorithms for decomposing given data into Gaussian beams, (2) develop new multiscale Gaussian beam methods for modeling acoustic, elastic and anisotropic waves, (3) analyze linearized inverse problems for acoustic and elastic wave equations by using multiscale Gaussian beams in the high frequency regime, (4) devise novel algorithms to implement the resulting linearized inversion formulas, and (5) validate the resulting algorithms by using synthetic data. Education objectives were to (1) incorporate research themes into lectures to engage students as active participants in the scientific discovery process, (2) train and enrich graduate and undergraduate students participating in the proposed research so as to develop human resources, and (3) increase minority awareness and participation in mathematical sciences. Intellectual merits of the proposed activities came from several aspects. (1) Our numerical methods will meet the significant scientific challenge in modeling acoustic, elastic and anisotropic waves in the high frequency regime. (2) Significant advances will be made in developing novel multiscale Gaussian-beam methods for seismic wave propagation. (3) A mathematical theory for Gaussian-beam based seismic imaging methods will be proposed by analyzing linearized asymptotic inversions with phase space transforms and Fourier integral operators. New multiscale Gaussian-beam based algorithms are developed for the first time for these applications. Broader impacts of this research rest upon both enormously potential scientific applications and educational activities. (1) The project has addressed several fundamental issues in scientific computing and applied mathematics, such as multiscale modeling and inverse problems, which are essential in geosciences and material sciences. Computational wave propagation has become a fundamental, vigorously growing technology in diverse science and engineering disciplines, ranging from seismic imaging, radar, sonar, medical imaging, remote sensing, submarine detection to nanotechnology. The new methodology resulting from multiscale Gaussian beam methods will have substantial impact on computational wave propagation and applications, particularly at high frequencies. Research results obtained through this research plan will be made readily available to those companies involved in geophysical explorations so that competitiveness of the U.S. oil industry will be enhanced. (2) The educational impact was significant in several areas. Graduate students have published scientific articles in peer reviewed journals and thus improved their overall scientific awareness, critical thinking, and communication skills. A new graduate course on "Computational Wave Propagation and Imaging" was developed, and the new advances made in the PIâ€™s research was incorporated into the course in a timely fashion. The impact of these combined efforts have strengthened the role of interactive research in graduate and undergraduate training at the Michigan State University (MSU) and expanded the opportunities for underserved minorities in science, mathematics and engineering. (3) Results obtained from the proposed activities were disseminated through publications and conference presentations to the communities of geophysical exploration and computational wave propagation.