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

Constant horizontal slowness-squared gradient model

Another medium that provides analytical time-to-depth conversion formulas is

$$w (z,x) = w_0 - 2 q_x x\;,$$ (20)

i.e., the slowness-squared changes linearly in the horizontal direction. In Appendix D, we show that in this case the coordinate mapping follows

$$x_0 (z,x) = \frac{2 w_0 x + q_x z^2}{w_0 + 2 q_x x + \sqrt{(w_0 - 2 q_x x)^2 - 4 q_x^2 z^2}}\;,$$ (21)

$$t_0 (z,x) = \frac{\sqrt{2} z \left[ 2 w_0 - 4 q_x x + \sqrt{... ...qrt{(w_0 - 2 q_x x)^2 - 4 q_x^2 z^2} \right]^{\frac{1}{2}}}\;.$$ (22)

The analytical expression for $v_d (t_0,x_0)$ is more complex and can be found in equations D-13 and D-14.

In Figure 8 we illustrate $x_0 (z,x)$ and $t_0 (z,x)$ in the model $w (z,x) = 1 - 0.052 x$ $\mbox{s}^2 / \mbox{km}^2$. To deal with the in-flow boundary issue, we apply the method described previously for the constant velocity gradient example. Unlike equation 16, equation 21 indicates varying geometrical spreadings in the domain. Figure 9 shows the corresponding analytical $Q^2 (z,x)$ and $v_d (t_0,x_0)$. The geometrical spreading is significant at the lower-right corner of the domain, which translates to the cost at approximately the same location in Figure 10. We use analytical Dix velocity as the input in the inversion. Starting from the Dix-inverted model and after three linearization updates, $E$ decreases to relative $0.0045\%$. The size of triangular smoother is $8$ m $\times$ $20$ m. The model misfit, as demonstrated in Figure 11, is also improved.

hs2analy
Figure 8. Analytical values of (top) $t_0$ and (bottom) $x_0$, overlaid with contour lines.

hs2grad
Figure 9. The (top) geometrical spreading and (bottom) Dix velocity associated with the model used in Figure 8.

hs2cost
Figure 10. The cost (top) before and (bottom) after inversion. The least-squares norm of cost $E$ is decreased from $10.431$ to $0.047$.

hs2error
Figure 11. The difference between exact model and (top) initial model and (bottom) inverted model. The least-squares norm of model misfit is decreased from $5.0\mbox{ km}^2/\mbox{s}^2$ to $0.5\mbox{ km}^2/\mbox{s}^2$.

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