Reflection-Migration

Reflection migration smears residuals along the yellow-colored ellipse in Figure 4.4e for a specified source and receiver pair. When two traces are migrated, Figure 4.3b suggests that the minimum width and height of the intersecting fat ellipses defines the resolution limits of reflection migration.

The formulas for migration resolution limits were more rigorously derived (Beylkin, 1985) by applying the migration kernel to traces that only contain the diffraction arrival from a single diffractor. For a localized scatterer^4.3in a background medium with smooth velocity variations, Beylkin showed that equation 4.6 asymptotically becomes the Fourier integral over the model wavenumbers $k_x$ and $k_z$ :

$$\delta m({\bf {x}})^{\textrm{mig}}=\alpha\!\!\int_{{{\mathcal D}}}\!\!e^{-i {\bf {k}}\cdot {\bf {x}}} \delta M({\bf {k}})\, J^{-1} dk_{x} dk_{z},$$ (4.9)

where $\alpha$ is related to geometrical spreading, $J$ is the Jacobian, which is derived in Appendix G, and the range of model wavenumbers $\mathcal D$ in the integral depends on the range of source-receiver pairs. In fact, the model wavenumber vector ${\bf {k}}$ can be equated to the sum of the source-scatterer and geophone-scatterer wavenumbers ${\bf {k}}={\bf {k}}_{gr_o}+{\bf {k}}_{sr_o}$ shown in Figure 4.5. I will now show how equations 4.6 and 4.9 can be used to estimate the resolution limits of the other wavepaths in Figure 4.4a-d.