Velocity-independent time-domain seismic imaging using local event slopes

Oriented prestack time migration

Mapping prestack data to the reflector position expressed in vertical traveltime is the task of prestack time migration. Analogously to dip moveout, prestack migration moves events not only across offsets but also across different midpoints provided that the reflectors are not flat. Using the oriented approach, mapping from the prestack domain $\{t,h,y\}$ to the time-migrated image domain $\{\tau,x\}$ is defined by the following equations derived in Appendix B:

$$\tau^2 = \frac{t\,p_h\,\left[(t-h\,p_h)^2 - h^2\,p_y^2\right]^2} {(t-h\,p_h)^2\,\left[t\,p_h + h\,(p_y^2-p_h^2)\right]}\;,$$ (18)

$$x = y - \frac{h\,t\,p_y}{t\,p_h + h\,(p_y^2-p_h^2)}\;.$$ (19)

Here $\{\tau,x\}$ are the time-migrated image coordinates that correspond to the vertical ray traveltime and location (Hubral, 1977).

Under the assumption of a hyperbolic diffraction moveout, the migration velocity becomes, analogously to equation 4, a data attribute completely defined by prestack slopes, as follows:

$${\frac{4}{v^2}} = {\frac{t\,\left[t\,p_h + h\,(p_y^2-p_h^2)\right]} {h\,(t-h\,p_h)}}\,.$$ (20)

Equations 18 and 20 transform to equations 3 and 4 when $p_y=0$, which corresponds to horizontal reflectors. Equations analogous to 18, 19, and 20 were derived previously by Billette et al. (2003) in a different way and in the depth imaging context.

Figures 11 and 12 show the output of prestack migration before and after stack. The result is comparable to that of conventional processing (Claerbout, 2005) but obtained two orders of magnitude faster, since each data point transforms directly to the image space through a one-to-one mapping instead of being spread along a wide migration impulse response, as in conventional prestack time migration.

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Figure 11. Output of oriented prestack time migration.

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Figure 12. Seismic image obtained by oriented prestack time migration.

Velocity-independent time-domain seismic imaging using local event slopes