Elastic wave-mode separation for VTI media

Simple model

I consider a 2D isotropic model characterized by the $V_{P}$ , $V_{S}$ and density shown in Figures 10(a)-fig:separate3-ro. The model contains negative P and S velocity anomalies that triplicate the wavefields. The source is located at the center of the model. Figure 11(a) shows the vertical and horizontal components of one snapshot of the simulated elastic wavefield (generated using the $8^{th}$ order finite difference solution of the elastic wave equation), Figure 11(b) shows the separation to P and S modes using $\nabla \cdot {}$ and $\nabla \times {}$ operators, and Figure 11(c) shows the mode separation obtained using the pseudo operators which are dependent on the medium parameters. A comparison of Figures 11(b) and fig:separate3-pI-f06 indicates that the $\nabla \cdot {}$ and $\nabla \times {}$ operators and the pseudo operators work identically well for this isotropic medium.

I then consider a 2D anisotropic model similar to the previous model shown in Figures 10(a)-fig:separate3-ro (with $V_{P}$ , $V_{S}$ representing the vertical P and S wave velocities), and additionally characterized by the parameters $\epsilon$ and $\delta$ shown in Figures 10(d) and fig:separate3-delta, respectively. The parameters $\epsilon$ and $\delta$ vary gradually from top to bottom and left to right, respectively. The upper left part of the medium is isotropic and the lower right part is highly anisotropic. Since the difference of $\epsilon$ and $\delta$ is great at the bottom part of the model, the qS waves in this region are severely triplicated due to this strong anisotropy.

Figure 12 illustrates the pseudo derivative operators obtained at different locations in the model defined by the intersections of $x$ coordinates 0.3, 0.6, 0.9 km and $z$ coordinates 0.3, 0.6, 0.9 km. Since the operators correspond to different combination of the parameters $\epsilon$ and $\delta$ , they have different forms. The isotropic operator at coordinates $x=0.3$  km and $z=0.3$  km, shown in Figure 12(a), is purely vertical and horizontal, while the anisotropic operators (Figure 12(b) to fig:separate3-aop22) have ``tails'' radiating from the center. The operators become larger at locations where the medium is more anisotropic, for example, at coordinates $x=0.9$  km and $z=0.9$  km.

Figure 13(a) shows the vertical and horizontal components of one snapshot of the simulated elastic anisotropic wavefield, Figure 13(b) shows the separation to qP and qS modes using conventional isotropic $\nabla \cdot {}$ and $\nabla \times {}$ operators, and Figure 13(c) shows the mode separation obtained using the pseudo operators constructed using the local medium parameters. A comparison of Figure 13(b) and 13(c) indicates that the spatially-varying derivative operators successfully separate the elastic wavefields into qP and qS modes, while the $\nabla \cdot {}$ and $\nabla \times {}$ operators only work in the isotropic region of the model.

aoppos,vs,ro,epsilon,delta
Figure 10. A $1.2$  km$\times$ $1.2$  km model with parameters (a) $V_{p0}=3$  km/s except for a low velocity Gaussian anomaly around $x=0.65$  km and $z=0.65$  km, (b) $V_{S0}=1.5$  km/s except for a low velocity Gaussian anomaly around $x=0.65$  km and $z=0.65$  km, (c) $\rho=1.0$  g/cm$^3$ in the top layer and $2.0$  g/cm$^3$ in the bottom layer, (d) $\epsilon$ smoothly varying from 0 to $0.25$ from top to bottom, (e) $\delta$ smoothly varying from 0 to $-0.29$ from left to right. A vertical point force source is located at $x=0.6$  km and $z=0.6$  km shown by the dot in panels (b), (c), (d), and (e). The dots in panel (a) correspond to the locations of the anisotropic operators shown in Figure 12 .

uI-f06,qI-f06,pI-f06
Figure 11. (a) One snapshot of the isotropic wavefield modeled with a vertical point force source at $x$ =0.6 km and $z$ =0.6 km for the model shown in Figure 10, (b) isotropic P and S wave modes separated using $\nabla \cdot {}$ and $\nabla \times {}$ , and (c) isotropic P and S wave modes separated using pseudo derivative operators. Both (b) and (c) show good separation results.

aop00,aop10,aop20,aop01,aop12,aop21,aop02,aop12,aop22
Figure 12. The $8^{th}$ order anisotropic pseudo derivative operators in the $z$ and $x$ directions at the intersections of $x$ =0.3, 0.6, 0.9 km and $z$ =0.3, 0.6, 0.9 km for the model shown in Figure 10.

uA-f06,qA-f06,pA-f06
Figure 13. (a) One snapshot of the anisotropic wavefield modeled with a vertical point force source at $x$ =0.6 km and $z$ =0.6 km for the model shown in Figure 10, (b) anisotropic qP and qS modes separated using $\nabla \cdot {}$ and $\nabla \times {}$ , and (c) anisotropic qP and qS modes separated using pseudo derivative operators. The separation of wavefields into qP and qS modes in (b) is not complete, which is obvious at places such as at coordinates $x=0.4$  km $z=0.9$  km. In contrast, the separation in (c) is much better, because the correct anisotropic derivative operators are used..

Elastic wave-mode separation for VTI media