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

Migration to zero offset

NMO correction accomplishes mapping to zero offset in the case of horizontal reflectors. Taking account of the reflector dip effect requires dip moveout (Hale, 1995). Combined with NMO, dip moveout maps the input data in time-offset-midpoint coordinates $\{t,h,y\}$ to time-midpoint coordinates $\{t_0,y_0\}$ at the zero-offset section. The corresponding oriented mapping, derived in Appendix B, is

$$t_0^2 = t\,\frac{\left[(t- h\,p_h)^2 - h^2\,p_y^2\right]^2} {(t-h\,p_h)^3}\;,$$ (16)

$$y_0 = y - \frac{h^2\,p_y}{t - h\,p_h}\;,$$ (17)

where $h = l/2$, and $p_h=\partial t/\partial h$ and $p_y=\partial t/\partial y$ are prestack slopes.

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Figure 7. A field 2-D dataset from the Gulf of Mexico used for numerical experiments. The display in this and other 3-D figures is composed of three sections from the cube indicated by vertical and horizontal lines.

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Figure 8. Prestack data slopes estimated from the dataset. a: Offset slope $p_h$. b: Midpoint slope $p_y$.

The full 2-D dataset and the estimated prestack slopes are shown in Figures 7 and 8, respectively. The output of migration to zero offset and stack is shown in Figures 9 (before stack) and 10 (after stack). All major events are properly transformed to the appropriate zero-offset positions.

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Figure 9. Output of oriented migration to zero offset.

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Figure 10. Seismic stack obtained by oriented migration to zero offset.

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