Hadamard-Babich ansatz and fast Huygens sweeping method for point-source 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 partial-differential-equation method is proposed to compute complex-valued eikonals in attenuating media.In Chapter 3, we develop the Hadamard-Babich (H-B) ansatz for frequency-domain point source elastic wave equations in an inhomogeneous medium in the high-frequency regime. First, we develop a novel asymptotic series, dubbed Hadamard’s ansatz, to form the fundamental solution of the Cauchy problem for the time-domain point-source (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 H-B ansatz for the frequency-domain point-source (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 Hadamard-Babich 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 high-frequency regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the Huygens-Kirchhoff secondary-source 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 matrix-vector products induced by the discretization of the Huygens-Kirchhoff 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 partial-differential-equation method to solve complex-valued eikonals in attenuating media. In the regime of high-frequency asymptotics, a complex-valued 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 real-space ray-tracing methods for complex-valued eikonals is proposed so that the real and imaginary parts of the eikonal function satisfy the classical real-space eikonal equation and a novel real-space advection equation, respectively, and we dub the resulting method the Eulerian partial-differential-equation method. We further develop highly efficient high-order methods to solve these two equations by using the factorization idea and the Lax-Friedrichs weighted essentially non-oscillatory (WENO) schemes. Numerical examples demonstrate that the proposed method yields highly accurate complex-valued eikonals, analogous to those from ray-tracing methods.