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The goal of wave-mode separation and wave-vector decomposition is to separate a full elastic wavefield into three wavefields with each corresponding to a different wave mode. This allows elastic reverse-time migration to handle each wave mode independently. Several of the previously proposed methods to accomplish this task require the knowledge of the polarisation vectors of all three wave modes in a given anisotropic medium. We propose a wave-vector decomposition method where the wavefield is decomposed in the wavenumber domain via the analytical decomposition operator with improved computational efficiency using low-rank approximations. The method is applicable for general heterogeneous anisotropic media. To apply the proposed method in low-symmetry anisotropic media such as orthorhombic, monoclinic, and triclinic, we define the two S modes by sorting them based on their phase velocities (S1 and S2), which are defined everywhere except at the singularities. The singularities can be located using an analytical condition derived from the exact phase-velocity expressions for S waves. This condition defines a weight function, which can be applied to attenuate the planar artefacts caused by the local discontinuity of polarisation vectors at the singularities. The amplitude information lost because of weighting can be recovered using the technique of local signal–noise orthogonalisation. Numerical examples show that the proposed approach provides an effective decomposition method for all wave modes in heterogeneous, strongly anisotropic media.