VTI migration velocity analysis using RTM

Joint inversion for two parameters

The tests in the previous sections show that we have a reliable objective function and successful inversion results for a single parameter. However, joint inversion for more than one parameter for each grid in the subsurface is far more challenging because of the ambiguity between parameters. As a result, the preconditioning scheme using geological and rock-physics information is crucial for its success.

In this test, we use the same synthetic data as in the last section. Unlike in the last example where we use the perfect velocity model, the starting models for velocity and $\epsilon$ are both inaccurate. The initial velocity model and $\epsilon$ model are shown in Figures 7(a) and 5(a), respectively. The angle gathers generated using these initial models are shown in Figure 7(b). Significant moveout in the angle-gather events indicates that the initial model is far from the true model. In fact, the initial velocity has a maximum of 15% error compared with the true velocity (Figure 1(a)), while the initial $\epsilon$ is about 50% smaller than the true value in the shallow part of the model. Notice that the error in velocity has a much larger effect on the kinematics of the seismic wave, hence a larger effect on the flatness in the angle domain.

After 40 iterations, we obtain the inverted velocity and $\epsilon$ models as shown in Figure 8(a) and 8(b). Comparing Figure 8(a) with Figure 1(a), we can conclude that the inversion has successfully recovered the high-resolution vertical structure in the shallow part of the model. Due to the limited illumination, the steep structure in the deeper part of the model is not well resolved. Comparing Figure 6(a) and Figure 8(b), we notice that, because of the error in velocity, the inversion does not converge to the same solution. This is an indication that we have not completely resolved the ambiguity between velocity and $\epsilon$ .

Angle gathers generated by the inverted model are shown in Figure 8(c). They are extracted from the same common-image points as in Figure 7(b). The improved model flattens the gathers across the whole section. Notice that the low-frequency energy in the water is the commonly seen wave-path energy for RTM images.

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Figure 7. Initial velocity model (a) in m/s and the angle gathers (b) obtained using initial velocity model. Initial $\epsilon$ model is shown in Figure 5(a). Model error causes significant curvatures in the angle gathers. Gathers are taken every 100 common image points from $x=0$ km to $x=9$ km.

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Figure 8. Inverted velocity model (a) in m/s and $\epsilon$ model (b) after 40 iterations. Angle gathers (c) obtained by the inverted model. Angle gathers are extracted at the same CIP as those in Figure 7(b). Improved velocity and $\epsilon$ flattens the corresponding angle gathers. Gathers are taken every 100 common image points from $x=0$ km to $x=9$ km.

VTI migration velocity analysis using RTM