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

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

The analytical expression for is more complex and can be found in equations D-13 and D-14.

In Figure 8 we illustrate and in the model . 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 and . 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, decreases to relative . The size of triangular smoother is m m. The model misfit, as demonstrated in Figure 11, is also improved.

hs2analy
Analytical values of (top)
and (bottom) , overlaid with contour lines.
Figure 8. |
---|

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

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

hs2error
The difference between exact
model and (top) initial model and (bottom) inverted model. The
least-squares norm of model misfit is decreased from
to
.
Figure 11. |
---|

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

2015-03-25