HadamardBabich ansatz and fast Huygens sweeping method for pointsource elastic wave equations in an inhomogeneous medium at high frequencies
Asymtotic methods are efficient for solving wave equations in the high frequency regime. In the thesis, we first develop a new asymptotic ansatz for point source elastic wave equations in an inhomogeneous medium. Then, we propose a fast Huygens sweeping method to construct a globally valid Green's functions in the presence of caustics. Finally, an Eulerian partialdifferentialequation method is proposed to compute complexvalued eikonals in attenuating media.In Chapter 3, we develop the HadamardBabich (HB) ansatz for frequencydomain point source elastic wave equations in an inhomogeneous medium in the highfrequency regime. First, we develop a novel asymptotic series, dubbed Hadamard’s ansatz, to form the fundamental solution of the Cauchy problem for the timedomain pointsource (TDPS) elastic wave equations in the region close to the source. Then, the governing equations for the unknown asymptotics of the ansatz are derived including the traveltime functions and dyadic coefficients. A matching condition is proposed to initialize the data of unknowns at the source. To treat singularity of dyadic coefficients at the source, smoother dyadic coefficients are then introduced. Directly taking the Fourier transform of Hadamard’s ansatz in time, we obtain the HB ansatz for the frequencydomain pointsource (FDPS) elastic wave equations. To verify the feasibility of the new ansatz, we truncate the ansatz to keep only the first two terms to compute the resulting asymptotic solutions. Numerical examples demonstrate the accuracy of our method.In Chapter 4, we propose a new truncated HadamardBabich ansatz based globally valid asymptotic method, dubbed the fast Huygens sweeping method, for computing Green's functions of FDPS elastic wave equations in inhomogeneous media in the highfrequency regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the HuygensKirchhoff secondarysource principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics can be treated automatically and implicitly. The precomputed asymptotic ingredients can be used to construct Green's functions of elastic wave equations for many different point sources and for arbitrary frequencies. The second novelty is that a butterfly algorithm is adapted to accelerate matrixvector products induced by the discretization of the HuygensKirchhoff integral. The computational cost of the butterfly algorithm is O(NlogN) which is in nearly optimal complexity in terms of the total number of mesh points N. The prefactor of the complexity depends only on the specified accuracy and is independent of the frequency parameter. Numerical examples are presented to demonstrate the performance and accuracy of the new method. In Chapter 5, we propose a Eulerian partialdifferentialequation method to solve complexvalued eikonals in attenuating media. In the regime of highfrequency asymptotics, a complexvalued eikonal is an essential ingredient for describing wave propagation in attenuating media, where the real and imaginary parts of the eikonal function capture dispersion effects and amplitude attenuation of seismic waves, respectively. Therefore, a unified framework to eulerianize several popular approximate realspace raytracing methods for complexvalued eikonals is proposed so that the real and imaginary parts of the eikonal function satisfy the classical realspace eikonal equation and a novel realspace advection equation, respectively, and we dub the resulting method the Eulerian partialdifferentialequation method. We further develop highly efficient highorder methods to solve these two equations by using the factorization idea and the LaxFriedrichs weighted essentially nonoscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complexvalued eikonals, analogous to those from raytracing methods.
Read
 In Collections

Electronic Theses & Dissertations
 Copyright Status
 AttributionNonCommercialShareAlike 4.0 International
 Material Type

Theses
 Authors

Song, Jian
 Thesis Advisors

Qian, Jianliang
 Committee Members

Zhou, Zhengfang
Christlieb, Andrew
Yang, Yang
 Date
 2022
 Subjects

Mathematics
 Program of Study

Applied Mathematics  Doctor of Philosophy
 Degree Level

Doctoral
 Language

English
 Pages
 206 pages
 Permalink
 https://doi.org/doi:10.25335/t19s3r95