Elastic wave-mode separation for TTI media

2D TTI fold model

Consider the 2D fold model shown in Figure 9. Panels 9(a)-fig:fold-aoppos show $V_{P0}$ , $V_{S0}$ , density, parameters $\epsilon$ , $\delta$ , and the local tilts $\nu$ of the model, respectively. The symmetry axis is orthogonal to the reflectors throughout the model. Figure 10 illustrates the separators obtained at different locations in the model and defined by the intersections of $x$ coordinates $0.15, 0.3, 0.45$  km and $z$ coordinates $0.15, 0.3, 0.45$  km, shown by the dots in Figure 9(f). Since the operators correspond to different combinations of the $V_{P0}/ V_{S0}$ ratio and parameters $\epsilon$ , $\delta$ , and tilt angle $\nu$ , they have different forms. However, the orientation of the operators conform to the corresponding tilts at the locations shown by the dots in Figure 9(f). For complex models, the symmetry axes vary spatially, which makes it difficult to rotate the wavefields to the local symmetry axis directions. Consequently, the elastic wavefields are reconstructed in untilted Cartesian coordinates, and when separating wave-modes, I use operators constructed in conventional Cartesian coordinates. To illustrate the relationship between the operators and the local tilts, the filters in Figure 10 are projected onto the local symmetry axes and the orthogonal directions at the filter location. As shown in Figure 4, the rotated filters (Figure ) show a clearer relation with the tilt angle, while the non-rotated filters (Figure ), which are used in the wave-mode separation, do not show a clear relation with the tilt angle.

Figure 11(a) shows the vertical and horizontal components of one snapshot of the simulated elastic anisotropic wavefield; Figure 11(b) shows the separation into P- and S-modes using divergence and curl operators; Figure 11(c) shows the separation into qP and qS modes using VTI filters, i.e., assuming zero tilt throughout the model; and Figure 11(d) shows the separation obtained with the TTI operators constructed using the local medium parameters with correct tilts. The isotropic separation shown in Figure 11(b) is incomplete; for example, at $x=0.4$ km and $z=0.1$ km, and at $x=0.4$ km and $z=0.35$ km, residuals for direct P and S arrivals are visible in the qP and qS panels, respectively. A comparison of Figures 11(c) and fig:pA indicates that the spatially-varying derivative operators with correct tilts successfully separate the elastic wavefields into qP and qS modes, while the VTI operators only work in the part of the model that is locally VTI.

vp,vs,ro,epsilon,delta,aoppos
Figure 9. A fold model with parameters (a) $V_{P0}$ , (b) $V_{S0}$ , (c) density, (d) $\epsilon$ , (e) $\delta$ , and (f) tilt angle $\nu$ . The dots in panel (f) correspond to the locations of the anisotropic operators shown in Figure 10.

rop
Figure 10. The TTI wave-mode separation filters projected to local symmetry axes and their orthogonal directions. Here, I use $\sigma =1$ in equation 12 to taper the polarization vector components before the Fourier transform. The filters correspond to the intersections of $x=0.15$ , $0.3$ , $0.45$  km and $z=0.15$ , $0.3$ , $0.45$  km for the model shown in Figure 9. The locations of these operators are also shown by the dots in Figure 9(f).

uA,pI,pV,pA
Figure 11. (a) A snapshot of the anisotropic wavefield simulated with a vertical point displacement source at $x=0.3$  km and $z=0.1$  km for the model shown in Figure 9. Panels (b) to (d) are the anisotropic qP and qS modes separated using isotropic, VTI, and TTI separators, respectively. The separation is incomplete in panels (b) and (c) where the model is strongly anisotropic and where the model tilt is large, respectively. Panel (d) shows the best separation among all.

Elastic wave-mode separation for TTI media