Biondi and Vaillant 2000 discuss the relative accuracy of offset plane-wave migration and CAM for wave-equation imaging. Both are derived from full downward continuation of 3-D prestack data with the Double Square Root (DSR) phase-shift operator:

(10) |

Offset plane wave (OPW) migration Mosher et al. (1997) performs migration of
each offset plane wave component of the data independently. It can be
interpreted as a reversed-order two-pass prestack migration, where an
initial cross-line zero-offset migration is followed by an in-line
prestack migration. Another interpretation in that the cross-line
offset wavenumber *k*_{hy} in equation (10) is set to zero
for downward continuation:

(11) |

Instead, for CAM, the cross-line offset wavenumber *k*_{hy} is
replaced in equation (10) by its stationary path
given in equation (9):

(12) |

Both migrations reduce the full 5-D phase-shift operator to a 4-D operator. In fact, when no multipathing occurs, only a 4-D slice of the 5-D wavefield contributes to the image.

Figures 9 and 10 show migration results. The final image cube has 4 dimensions, the last being the common-image gather (CIG) ray parameter axis, generated by slant stack Prucha et al. (1999). The CIGs are flat for the first reflectors (dips ) in both images. For OPW migration, non-flat gathers start at dip , whereas only a dip of causes trouble to CAM.

Figure 9

Figure 10

In order to have a reference for comparison, we migrated the 4-D common-azimuth data with 5-D phase-shift migration, after zero-padding along a fictitious cross-line offset axis (Figure 8). Figure 11 shows flat gathers even for the most strongly dipping reflectors.

For using the full 5-D phase-shift operator, we added a fictitious
cross-line offset axis by zero-padding common-azimuth data. The
``arbitrary'' parameters *n*_{hy} and *d*_{hy} are chosen in order to
avoid wraparound problems in Fourier Transforms (*n*_{hy} large
enough) and to have the exact value of *k*_{hy} included in our
*k*_{hy} range. In practice, we used *n*_{hy}=24 and *d*_{hy}=50*m*.

Figure 11

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