Elastic wave-vector decomposition in heterogeneous anisotropic media

Discussion

The proposed wave-vector decomposition method utilizes an analytical expression for locating singularities (equation 25) as the basis for a non-stationary smoothing operator (equation 27) defined as a weighting in the wavenumber domain. The analytical expression is a function of the components of the Christoffel matrix $\mathbf{G}$ variable in space and phase directions defined by the wavenumbers. Because the S-wave phase velocities in low-symmetry media (including orthorhombic, monoclinic, and triclinic) can be computed based on the same generic formulas (equations 21 and 22) with a corresponding change in $\mathbf{G}$ , the same analytical condition for locating singularities can also be used in those media. In this paper, we have demonstrated the applicability of the proposed method for wave-vector decomposition in low-symmetry anisotropic media using synthetic examples with orthorhombic and triclinic symmetries.

Alternatively to solving the Christoffel equation numerically for exact values of polarizations, one may choose to use an analytical approximation in weakly anisotropic media derived from perturbation theory (Farra and Pšencík, 2003). This choice may lead to increased computational efficiency in complex models.

Generally, the knowledge of polarization vectors and their applicability are based on the underlying assumption in which the medium is assumed to be locally homogeneous relative to the propagating frequency of the waves. In the case of a larger degree of heterogeneity such as strong contrasts and considerable velocity gradients, this assumption is approximate and may need special care.

To implement wave-vector decomposition, the proposed method uses the low-rank approximations for the decomposition operator (equation 12). This allows us to avoid explicitly computing and storing the polarizations of the mode of interest at every grid point, which would be prohibitively expensive. The proposed method is appropriate for decomposing the elastic wavefields during the backward propagation step in elastic reverse-time migration (RTM) (Wang et al., 2016) and full-waveform inversion (FWI) (Wang et al., 2015). We did not consider the problem of separating wave modes in recorded surface seismic data.

Elastic wave-vector decomposition in heterogeneous anisotropic media