Numerical Examples

To validate the proposed TTI decoupled equations, several synthetic modeling and RTM examples are demonstrated. For comparison, results from the TTI coupled equations (equation 2 in Fletcher et al. (2009)) are also presented.

First, I demonstrate the proposed algorithm using an impulse response. Figure 3.2 shows time snapshots of wave propagation at time $t=1~s$ in a 2D homogeneous TTI medium with $V_{p_0}=3000~m/s$ , $\epsilon =0.24$ , $\delta =0.1$ and $\theta =45^\circ$ . By solving the TTI coupled equations with $V_{s_0}$ set to zero, Figure 3.2a is obtained where we can clearly see an elliptical P wavefront as well as a diamond-shape SV wavefront. Figure 3.2b shows a same time snapshot as that in Figure 3.2a, but with a nonzero SV wave velocity ( $V_{s_0}=V_{p_0}/2$ ). By adding a finite SV wave velocity, SV wave triplications are successfully removed but the shear wave, which is considered as an artifact in P wave modeling and migration, is still present in Figure 3.2b. Figure 3.2c and 3.2d demonstrate the separated P wave and SV wave snapshots from the TTI decoupled equations. Compared to Figure 3.2b, it is obvious that Figure 3.2c only has the P wave component and Figure 3.2d only contains the SV wave component. Therefore, the proposed TTI decoupled equations offer a better modeling and migration alternative since the P wave and SV wave modes are separated. Since the SV wave is usually considered as an artifact in conventional P wave modeling and migration, I only focus on the TTI decoupled P wave equation in the following discussion.

Figure: Wavefield snapshots at time $t=1~s$ in a homogeneous TTI medium ( $V_{p_0}=3000~m/s$ , $\epsilon =0.24$ , $\delta =0.1$ , $\theta =45^\circ$ ) computed using different TTI equations. (a) is generated from the TTI coupled equations with $V_{s_0}=0$ . (b) is the same as (a) but with a nonzero SV wave velocity. (c) and (d) are the results from the solution of the TTI decoupled P and SV wave equations, respectively.

Second, I show some modeling examples using a 2D inhomogeneous wedge model where sharp contrasts exist in all parameter models (Figure 3.3). The model shown here is the same as the one used by Duveneck and Bakker (2011), which is originally designed to compare abilities of different approaches that deal with the instability problem in TTI media. To avoid SV waves generated from the source, they placed an isotropic layer on the top. I did so as to compare, but it is not necessary for my algorithm. Figure 3.4a shows the interface in sharp contrast of this anisotropic model. Figure 3.4b and Figure 3.4c are snapshots at time $t=1~s$ computed using the TTI coupled equations with $V_{s_0}=0$ and $V_{s_0}=\sqrt{(\epsilon-\delta)V^2_{p_0}/0.75}$ , respectively. SV wavefront appears to come out from the top interface of the wedge. Using the decoupled P wave equation, I get a clean wavefield snapshot without any shear wave contamination (Figure 3.4d). Figure 3.5 compares the same snapshots at time $t=1.5~s$ . Notice that the instabilities start to appear in Figure 3.5b for the $V_{s_0}=0$ case. Adding a nonzero shear wave velocity can mitigate the instability problem and thus stabilize the wave propagation (Figure 3.5c), but SV wavefronts are clearly present. Figure 3.5d from the TTI decoupled P wave equation solution is not only stable but also much cleaner than Figure 3.5c since the equation does not contain any shear wave components. Figure 3.6 displays wavefield snapshots at a later time ($t=4~s$ ). Even with a nonzero shear wave velocity, the wave propagation becomes unstable for long propagation times (Figure 3.6c). However, the decoupled P wave equation proceeds with a stable response and the wavefield exits the computation grid without ever becoming unstable (Figure 3.6d).

Figure 3.3: Anisotropic parameter values of a 2D wedge model used in TTI modeling tests: (a) $V_{p_0}$ , (b) $\theta$ , (c) $\epsilon$ and (d) $\delta$ .

Figure: Wavefield snapshots of the wedge model at time $t=1~s$ . (a) shows the interface with sharp dip contrasts. (b) The wavefield snapshot from the solution of the TTI coupled equations with $V_{s_0}=0$ . (c) Same as (b) but with a non-zero SV wave velocity. (d) is computed by using the TTI decoupled P wave equation.

Figure: The same plot as Figure 3.4 but the wavefield snapshots are recorded at time $t=1.5~s$ . (b) starts to blow up while (c) and (d) keep stable.

Figure: Wavefield snapshots at time $t=4~s$ . The instabilities occur in both (b) and (c). But wave propagation in (d) remains stable.

To further verify the proposed algorithm, I implement an anisotropic RTM using the TTI decoupled P wave equation. The BP 2007 TTI dataset is used in my RTM test. Figure 3.7 shows the parameter values for $V_{p_0}$ , $\epsilon$ , $\delta$ and $\theta$ of a small region of the 2D BP TTI model. Rapid variation of the tilt angle shown here presents challenges to TTI RTM. Before doing migration, a modeling test showing the instability problem is demonstrated. Figure 3.8 displays forward modeling snapshots at time $t=4~s$ . Notice that numerical instabilities arise in Figure 3.8b from the place where rapid dip variations exist (see Figure 3.8a). Using a finite shear wave velocity ( $V_{s_0}=\sqrt{(\epsilon-\delta)V^2_{p_0}/0.75}$ ) can stabilize wave propagation and the resulting snapshot is displayed in Figure 3.8c. Notice that there are some visible SV wave artifacts present in Figure 3.8c. Instead of using the TTI coupled equation with a nonzero shear wave velocity, I use the TTI decoupled P wave equation given by equation 3.11 to propagate the wavefield and the result is shown in Figure 3.8d. The wavefield is as stable as Figure 3.8c but is clean due to the complete isolation of the SV wave component in the P wave equation. The TTI decoupled P wave equation provides a stable result and is completely independent of variations in the anisotropic parameter models. In addition, it is absolutely free of any SV wave components.

Figure 3.7: Partial region of the 2D BP TTI model: (a) $V_{p_0}$ , (b) $\theta$ , (c) $\epsilon$ and (d) $\delta$ .

Figure 3.8: Wavefield snapshots of the BP 2D TTI model. (a) The gradient of dip angle model (displayed with an appropriate clip value) where black line segments correspond to sharp dip contrasts in the dip model. (b) The wavefield snapshot from the TTI coupled equations with $V_{s_0}=0$ . The instability problem arises from where high dip contrasts exist. (c) The same as (b) but with a finite $V_{s_0}$ . The wavefield blow-up disappeared. (d) The wavefield snapshot computed using the TTI decoupled P wave equation.

Finally, RTM results of the corresponding region are presented. Figure 3.9 compares VTI and TTI reverse time migration. Figure 3.9a is obtained using the VTI decoupled P wave equation described in equations 3.4. Due to ignoring the tilt angle, imaging of dipping events is severely affected and they seem to be mispositioned and defocused in the VTI RTM image. Using the TTI decoupled P wave equation given by equations 3.11, a TTI RTM image is achieved and the resulting image is shown in Figure 3.9b. It is clear that events are well positioned and focused at interface boundaries in the TTI RTM image using the TTI decoupled P wave equation. This is further demonstrated in Figure 3.10 which shows zoom views of Figure 3.9a and 3.9b.

Figure 3.9: Comparison of (a) VTI RTM and (b) TTI RTM images of the partial BP model.

Figure 3.10: Zoom view of Figure 3.9. (a) $V_{p_0}$ , (b) VTI RTM image and (c) TTI RTM image.