Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media

BP 2007 TTI model

The last example displays the application to the BP 2007 TTI model (Figure 11). Vertical qS-wave velocity is set to equal sixty percent of the vertical qP-wave velocity everywhere. Extrapolation steps of $\Delta{x}=\Delta{z}=12.5$ m and $\Delta{t}=1.0$ ms are used here. Because the principal axes of the medium are not aligned with the Cartesian axes, we have apply the Bond transformation to get the stiffness matrix under the Cartesian system. Before wavefield extrapolation, separated representations of the mixed operator matrixes are constructed using the low-rank decomposition approach within the computational zone. For this complex model, the ranks are about $30$ for the decomposition of the involved matrixes. As shown in Figure 12, the approach describes very well the propagations of the decoupled qP- and qS-waves as well as the total displacement fields even for this complex TTI model. We can clearly observe the converted waves from the dipping salt flanks and other strong-contrast interfaces. And the qP- and qS-waves are free of numerical dispersion in the decoupled and total wavefields.

vp0,epsi,del,the
Figure 11. Partial of BP 2007 TTI model with parameters of (a) vertical P-wave velocity, Thomsen coefficients (b) $\epsilon$ and (c) $\delta$ , and (d) tilt angle $\theta$ .

ElasticPxKS,ElasticPzKS,ElasticSxKS,ElasticSzKS,ElasticxKS,ElasticzKS
Figure 12. Synthesized decoupled and total displacement fields at the time of 1.2 s using the low-rank pseudo-spectral solution with the $k$ -space adjustment that simultaneously extrapolate and decouple qP- and qSV-wave fields in the BP 2007 TI model: (a) x- and (b) z-components of qP-wave fields; (c) x- and (d) z-components of qSV-wave fields; (e) x- and (f) z-components of the total displacement fields.

Simulating propagation of decoupled elastic waves using low-rank approximate mixed-domain integral operators for anisotropic media