The existence of Gaussian beam solution to hyperbolic PDEs has been known to the pure mathematics community since sometime in the 1960s [3]. It enjoys popularity afterwards due to its ability to resolve the caustics problem and its efficiency [49, 28, 31]. In this thesis, we will focus on the extension of the multi-scale Gaussian beam method and its application to seismic wave modeling and inversion. In the first part of thesis, we discuss the application of the multi-scale Gaussian beam method to the inverse problem. A new multi-scale Gaussian beam method is introduced for carrying out true-amplitude prestack migration of acoustic waves. After applying the Born approximation, the migration process is considered as shooting two beams simultaneously from the subsurface point which we want to image. The Multi-scale Gaussian Wavepacket transform provides an efficient and accurate way for both decomposing the perturbation field and initializing Gaussian beam solution. Moreover, we can prescribe both the region of imaging and the range of dipping angles by shooting beams from a subsurface point in the region of imaging. We prove the imaging condition equation rigorously and conduct error analysis. Some numerical approximations are derived to improve the efficiency further. Numerical results in the two-dimensional space demonstrate the performance of the proposed migration algorithm. In the second part of thesis, we propose a new multiscale Gaussian beam method with reinitialization to solve the elastic wave equation in the high frequency regime with different boundary conditions. A novel multiscale transform is proposed to decompose any arbitrary vector-valued function to multiple Gaussian wavepackets with various resolution. After the step of initializing, we derive various rules corresponding to different types of reflection cases. To improve the efficiency and accuracy, we develop a new reinitialization strategy based on the stationary phase approximation method to sharpen each single beam ansatz. This is especially useful and necessary in some reflection cases. Numerical examples with various parameters demonstrate the correctness and robustness of the whole method. There are two boundary conditions considered here, the periodic and the Dirichlet boundary condition. In the end, we show that the convergence rate of the proposed multiscale Gaussian beam method follows the convergence rate of the classical Gaussian beam solution.
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