A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations

Spiral model

vz0
Figure 12. (Top) a synthetic model and (middle) Dix velocity converted to depth. Both overlaid with image rays. (Bottom) the model perturbation for testing linearization.

Figure 12 shows a synthetic model borrowed from Cameron et al. (2008). The Dix inversion recovers the shallow part of the model but deteriorates quickly as geometrical spreading of image rays grows in the deeper section.

As a simple verification for the linearization process, we add a small positive velocity perturbation at location $(1,3)$ km to the synthetic model. Comparisons between the exact and linearly predicted attributes are illustrated in Figures 13, 14 and 15. In accordance with forward modeling, where we solve firstly $t_0$, then $x_0$, and finally $f$, the linearization (see Appendix B) is carried out following the same sequence. First, Figure 13 justifies our upwind finite-differences implementation of the linearized eikonal equation. The positive perturbation in $v$ in Figure 12 causes $t_0$ to decrease in a narrow downwind region. Next, the area affected by the perturbation in Figure 14 is wider than that in Figure 13. It also has both positive and negative amplitudes. These phenomenon are physical because image rays should bend in opposite directions in response to the perturbation. Finally, effects in cost $f$ in Figure 15 show alternating polarities and are broader in width compared to that of $d t_0$ and $d x_0$. They indicate a complicated dependency of $f$ on $w$. Note the good agreements in both shape and magnitude between exact and linearly predicted quantities in all three steps.

Because there is no analytical formula for Dix velocity in this model, we compute $v_d$ by tracing image rays numerically in the exact model $v$. Also, based on Figure 12, there is no in-flow boundary other than $z =0$. Therefore, we do not need to extend the domain as in the preceding examples. We use the Dix-inverted model as the prior model and run the inversion. It turns out that the first linearization update is sufficient for achieving the desired global minimum as shown in Figure 16.

pdt
Figure 13. (Top) exact $d t_0$ and (bottom) linearly predicted $d t_0$ by equation B-4.

pdx
Figure 14. (Top) exact $d x_0$ and (bottom) linearly predicted $d x_0$ by equation B-3.

diffcost
Figure 15. (Top) exact $d f$ and (bottom) linearly predicted $d f$ by equation B-5.

bgrad
Figure 16. (Top) the exact $\delta w$ and (middle) the computed $\delta w$ of the first linearization step. (Bottom) the inverted interval velocity model. Compare with Figure 12.

A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations