Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media

BP 2007 TTI model

This example displays the wave mode separation and vector decomposition results in the BP 2007 TTI model (Figure 9). A point-source is placed near the second salt body at the location of (35.625, 5.0) km. Before wavefield extrapolation, separated representations of the operator matrixes are constructed using the low-rank decomposition approach within the computational zone. It took 217.0 seconds to decompose the operator matrixes for mode separation with rank $N, M\in[15,17]$ , and 345.5 seconds to decompose the operator matrixes for vector decomposition with rank $N, M\in[16,18]$ . For one time step, it took 4.3, 16.0 and 34.8 seconds to extrapolated, separate and decompose the elastic wavefields, repectively. From the separated and decomposed wavefields (Figure 10), we can clearly observe the converted waves from the dipping salt flanks. Due to the low velocities of qSV-wave in some directions at some locations, there are numerical dispersion in the qSV-wave fields. In spite of the dispersion, we obtain well separated qP- and qSV-wave fields, as well as their decomposed x- and z-components.

Then we investigate the effect of the relative accuracy requirement on wave mode separation. Figure 11 demonstrates the separated P- and qSV-wave fields, and their variations when we relax the approximation level from $10^{-6}$ to $10^{-3}$ . It took 174.0 seconds to decompose the operator matrixes for mode separation with rank $N, M\in[7,8]$ , and 345.5 seconds to decompose the operator matrixes for vector decomposition with rank $N, M\in[8,9]$ . And it took 8.0 and 14.9 seconds to separate and decompose the elastic wavefields, repectively. The results are acceptable although more errors are introduced in the separated wavefields when we turn down the relative accuracy requirement.

To further analysize the rough relationship of rank $(N,M)$ with the model complexity, we smooth the BP TTI model by applying a 2D triangle smoothing operator with the radius of 1875m on both x- and z-axes (Figure 12). To maintain the range of the tilt angles, we first double the values of the original model and then apply the smoothing operation for this parameter. Figure 13 demonstrates the synthetic elastic wavefields and the mode separation and vector decomposition results. In this case, it took 207.0 seconds to decompose the operator matrixes for mode separation with rank $N, M\in[13,14]$ , and 310.2 seconds to decompose the operator matrixes for vector decomposition with rank $N, M\in[14,16]$ . It took 15.0 and 29.4 seconds to separate and decompose the elastic wavefields, repectively. We observe that the ranks further decrease to about 12 if we double the smoothing radius to 3750m. For homogeneous TI medium, the ranks automatically decrease to 1. We obtain accurate mode separation and decomposition of the isotropic and elastic wavefields at negligible computational cost with rank $N=M=1$ , if $\epsilon$ , $\delta$ and $\theta$ are all set as $0.0$ in the models.

vp0,epsi,del,the
Figure 9. BP 2007 TTI model with parameters of (a) vertical P-wave velocity, Thomsen coefficients (b) $\epsilon$ and (c) $\delta$ , and (d) tilt angle $\theta$ .

Elasticx,Elasticz,ElasticSepP,ElasticSepSV,ElasticPx,ElasticPz,ElasticSVx,ElasticSVz
Figure 10. Mode separation and vector decomposition using low-rank approximate algorithms in the BP 2007 TTI model: (a) x- and (b) z-components of the synthetic elastic displacement wavefields at 1.4s; (c) and (d) are the separated scalar qP- and qSV-wave fields; (e) x- and (f) z-components of vector qP-wave fields; (g) x- and (h) z-components of vector qSV-wave fields.

ElasticSepP3,ElasticSepSV3,ElasticSepP6vs3Dif,ElasticSepSV6vs3Dif
Figure 11. Elastic wave mode separation using low-rank approximation with relaxed accuracy requirements: Separated (a) qP- and (b) qSV-wave fields at the error level of $10^{-3}$ in low-rank decomposition; Differences of (c) qP- and (d) qSV-wave fields to those separated with the error level of $10^{-6}$ .

vp0,epsi,del,the
Figure 12. Smoothed BP 2007 TTI model with parameters of (a) vertical P-wave velocity, Thomsen coefficients (b) $\epsilon$ and (c) $\delta$ , and (d) tilt angle $\theta$ . 2D triangle smoothing with the smoothing radius of 1875m on both axis is applied to the paramaters shown in Figure 9.

Elasticx,Elasticz,ElasticSepP,ElasticSepSV,ElasticPx,ElasticPz,ElasticSVx,ElasticSVz
Figure 13. Mode separation and vector decomposition using low-rank approximate algorithms in the BP 2007 TTI model: (a) x- and (b) z-components of the synthetic elastic displacement wavefields at 1.4s; (c) and (d) are the separated scalar qP- and qSV-wave fields; (e) x- and (f) z-components of vector qP-wave fields; (g) x- and (h) z-components of vector qSV-wave fields.

Fast algorithms for elastic-wave-mode separation and vector decomposition using low-rank approximation for anisotropic media