Elastic wave-mode separation for VTI media

Sigsbee model

My second model (Figure 14) uses an elastic anisotropic version of the Sigsbee 2A model (Paffenholz et al., 2002). In the modified model, $V_{P0}$ is taken from the original model, the $V_{P0}/V_{S0}$ ratio ranges from $1.5$ to $2$ , the parameter $\epsilon$ ranges from 0 to $0.48$ (Figure 14(d)) and the parameter $\delta$ ranges 0 from to $0.10$ (Figure 14(e)). The model is isotropic in the salt and the top part of the model. A vertical point force source is located at coordinates $x=14.5$  km and $z=5.3$  km to simulate the elastic anisotropic wavefield.

Figure  shows one snapshot of the modeled elastic anisotropic wavefields using the model shown in Figure 14. Figure  illustrates the separation of the anisotropic elastic wavefields using the $\nabla \cdot {}$ and $\nabla \times {}$ operators, and Figure  illustrates the separation using my pseudo derivative operators. Figure  shows the residual of unseparated P and S wave modes, such as at coordinates $x=13$  km and $z=7$  km in the qP panel and at $x=11$  km and $z=7$  km in the qS panel. The residual of S waves in the qP panel of Figure  is very significant because of strong reflections from the salt bottom. This extensive residual can be harmful to under-salt elastic or even acoustic migration, if not removed completely. In contrast, Figure  shows the qP and qS modes better separated, demonstrating the effectiveness of the anisotropic pseudo derivative operators constructed using the local medium parameters. These wavefields composed of well separated qP and qS modes are essential to producing clean seismic images.

In order to test the separation with a homogeneous assumption of anisotropy in the model, I show in Figure  the separation with $\epsilon =0.3$ and $\delta =0.1$ in the $k$ domain. This separation assumes a model with homogeneous anisotropy. The separation shows that there is still residual in the separated panels. Although the residual is much weaker compared to separating using an isotropic model, it is still visible at locations such as at coordinates $x=13$  km and $z=7$  km, and $x=13$  km and $z=4$  km in the qP panel and at $x=16$  km and $z=2.5$  km in the qS panel.

vp,vs,ro,epsilon,delta
Figure 14. A Sigsbee 2A model in which (a) is the P wave velocity (taken from the original Sigsbee 2A model (Paffenholz et al., 2002) ), (b) is the S wave velocity, where $V_{P0}/V_{S0}$ ratio ranges from $1.5$ to $2.0$ , (c) is the density ranging from $1.0$  g/cm$^3$ to $2.2$  g/cm$^3$ , (d) is the parameter $\epsilon$ ranging from $0.20$ to $0.48$ , and (e) is the parameter $\delta$ ranging from 0 to $0.10$ in the rest of the model.

uA-f24-wom
Figure 15. Anisotropic wavefield modeled with a vertical point force source at $x=14.3$  km and $z=5.3$  km for the model shown in Figure 14.

qA-f24-wom
Figure 16. Anisotropic qP and qS modes separated using $\nabla \cdot {}$ and $\nabla \times {}$ for the vertical and horizontal components of the elastic wavefields shown in Figure . Residuals are obvious at places such as at coordinates $x=13$  km and $z=7$  km in the qP panel and at $x=11$  km and $z=7$  km in the qS panel.

pA-f24-wom
Figure 17. Anisotropic qP and qS modes separated using pseudo derivative operators for the vertical and horizontal components of the elastic wavefields shown in Figure . They show better separation of qP and qS modes.

mA-f24-wom
Figure 18. Anisotropic qP and qS modes separated in the $k$ domain for the vertical and horizontal components of the elastic wavefields shown in Figure . The separation assumes $\epsilon =0.3$ and $\delta =0.1$ throughout the model. The separation is incomplete. Residuals are still visible at places such at coordinates $x=13$  km and $z=7$  km, and $x=13$  km and $z=4$  km in the qP panel and at $x=16$  km and $z=2.5$  km in the qS panel.

Elastic wave-mode separation for VTI media