A $k$ -Space Pseudospectral Method for Elastic Wave Propagation in Heterogeneous Anisotropic Media

This paper presents the theory of the k-space method generalized to model elastic wave propagation in heterogeneous anisotropic media. The k-space methods are promising time integration techniques giving, in conjunction with collocation spectral methods, accurate and efficient numerical schemes for problems in heterogeneous media. In this paper, the k-space operator is derived in a spatially continuous form using the Fourier analysis of the displacement formalism of elastodynamics. An efficient numerical algorithm is then constructed by applying a Fourier collocation spectral method, leading to define the discrete k-space scheme. The proposed method is temporally exact for homogeneous media, unconditionally stable for heterogeneous media, and also allows larger time steps without loss of accuracy. Implementation of the method is discussed in detail. The method is validated through a set of numerical tests. The numerical results show the efficacy of the method compared with the conventional schemes.