Modeling elastic waves in heterogeneous anisotropic media using a k-space method

We generalize the theory of the k-space method to the case of elastic wave propagation in heterogeneous anisotropic media. The k-space operator is derived in the spatially continuous form using the displacement formalism of elastodynamics. The k-space scheme is then discretized in space using a Fourier collocation spectral method. This leads to an efficient and accurate numerical algorithm, where the time advancement can be performed in order of N operations, where N is the number of unknowns. As opposed to the classical k-space theory for the elastic waves in isotropic media [1], [2], the new algorithm does not need any field splitting. Hence, it is more efficient once used to model isotropy. The proposed method is temporally exact for homogeneous media, unconditionally stable for heterogeneous media, and also allows larger time-steps without loss of accuracy. We validate the method against canonical model problems of elastodynamics.