Moveout-based wave-equation migration velocity analysis |

Note that in order to handle multiple events, we use a local window of length along the depth axis. For the rest of the paper, the integration bound for variable is always , and each is defined within that window around image point (x,z). represents a windowed version of the entire image.

This methodology is borrowed from Luo and Schuster (1991) who tried to find the relation of the travel-time perturbation to the slowness change. Then can be found using the rule of partial derivatvies for implicit functions (please refer to the appendix A):

in which . and indicate the first and second derivatives in (depth). In practice, eq. (2) will be greatly simplified if we evaluate this expression at , in other words . In fact, this will always be the case if we update the intial slowness after each iteration. The simplified relation becomes

and the term is indeed the wave-equation image-space tomographic operator. Each part in eq. (3) has clear physical implications: the term acts as an energy term to normalize the amplitude of the back-projected image; the back-projected image, is built based on the initial image; it also has a first-order derivative that introduces a proper phase shift, ensuring a well behaved slowness update from the tomographic operator.

sensKer1w.zref,sensKer1w.Tilt.zref
Slowness sensitivity kernel at incident angle
for a flat reflector (a) and a dipping reflector (b).
Figure 1. |
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For a simple illustration of eq. (3), the slowness sensitivity kernel is calculated, by back-projecting a shift perturbation that has one single spike at . A uniform background velocity of 2000 m/s is used. Figure 1(a) shows the sensitivity kernel if the reflector is flat, and figure 1(b) shows the sensitivity kernel with a dipping reflector (dip angle = ). As is clearly shown in these two plots, this operator will project the slowness perturbation along the corresponding wave path based on the location and reflection angle of the image shift.

Moveout-based wave-equation migration velocity analysis |

2011-05-24