Hadamard-Babich Ansatz For Point-Source Elastic Wave Equations In Variable Media At High Frequencies

Starting from Hadamard's method, we develop Babich's ansatz for the 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 elastic wave equations in the region close to the source. Using the properties of generalized functions, we derive governing equations for the unknown asymptotics of the ansatz including the travel time functions and dyadic coefficients. In order to derive the initial data of the unknowns at the point source, we further propose a condition for matching Hadamard's ansatz with the homogeneous-medium fundamental solution at the point source. To treat singularity of dyadic coefficients at the source, we then introduce smoother dyadic coefficients. Directly taking the Fourier transform of Hadamard's ansatz in time, we obtain a new ansatz, dubbed Hadamard--Babich ansatz, for the frequency-domain point-source elastic wave equations. To verify the feasibility of the new ansatz, we truncate the ansatz to keep only the first two terms, and we further develop partial-differential-equation--based Eulerian approaches to compute the resulting asymptotic solutions. Numerical examples demonstrate the accuracy of our method.