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

In a medium with a constant velocity gradient

where . The migration velocity and Dix velocity take the following expressions:

It is easy to verify from equations 16 and 17 that , and . Because there is no geometrical spreading of image rays in this case, the Dix velocity will be equal to the interval velocity according to equation 3. However, a Dix-inverted model will still be distorted if because of the lateral shift of image rays.

Figures 3 and 4 show a velocity model with km/s and the corresponding analytical , and . Clearly, the right domain boundary is of in-flow type that violates our assumption. To address this challenge, we include Dix velocity in regions beyond the original left and right boundaries during inversion, but mask out the cost in these regions. It means that the time-to-depth conversion is performed in a sub-domain of time-domain attributes, such that information on the in-flow boundary is available. Afterwards, we apply Dix inversion to the expanded model and use the result as the prior model. We use the exact Dix velocity in equation 19 for evaluating the right-hand side of 5. Then, in total three linearization updates are carried out, which decreases to relative . The radiuses of triangular smoother in shaping are m vertically and m horizontally ( m m). At last, we cut the computational domain back to its original size. Figures 5 and 6 compare the cost and model misfit before and after inversion.

We also synthesize data with Kirchhoff modeling (Haddon and Buchen, 1981) for several horizontal reflectors using the exact model, and examine the subsurface scattering-angle common-image-gathers from Kirchhoff prestack depth migration (Xu et al., 2001) as an evidence of interval velocity improvements. In Figure 7, the shallower events do not improve significantly because the image rays have not yet bent considerably. Deeper events become noticeably flatter after applying the proposed method.

vgrad
(Top) a constant velocity gradient
model and (bottom) the analytical Dix velocity . A curved image
ray is mapped to the time domain as a straight line.
Figure 3. |
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analy
Analytical values of (top)
and (bottom) of the model in Figure 3. Both figures
are overlaid with contour lines that, according to equation
7, are perpendicular to each other. Each contour line of
is an image ray, while the contours of illustrate the
propagation of a plane-wave.
Figure 4. |
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cost
The cost defined by equation
9 (top) before and (bottom) after inversion. The
least-squares norm of cost is decreased from to .
Figure 5. |
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error
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 6. |
---|

cigv
Comparison of the subsurface
scattering-angle common-image-gathers at km of (left)
exact model, (middle) prior model and (right) inverted model.
Figure 7. |
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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