Depth focusing analysis is a methodology for velocity estimation based
on the fact that the total wave propagation time is invariant: the
total traveltime from the real reflection point obtained with the
real propagation velocity is equivalent to the total traveltime
obtained from the focus reflection image obtained with a wrong
propagation migration velocity plus a residual time related with the
error in velocity. This residual time, in a horizontal stratified
media case with constant velocity *V*_{r} (see Figure 2), is equal to
a vertical time associated with a depth error a paraxial approximation MacKay and Abma (1992):

(6) |

Figure 2

Figure 3 presents the migration of a dipping reflector with the media
velocity and with a wrong velocity *V*_{m}. Keeping the notation for
the migration velocity *V*_{m}, real velocity *V*_{r}, and the
downward continued point with the migration velocity to the imaging
condition *m*=(*x*_{m},*y*_{m},*Z*_{m}), the real position of the dipping
reflector point is *r*=(*x*_{r},*y*_{r},*z*_{r}) and the focusing point
is *f*=(*x*_{f},*y*_{f},*Z*_{f}). The total time *t*_{T} can be related
to the focus time *t*_{f} by the following equation
MacKay and Abma (1992):

(7) |

(8) |

(9) |

The focusing analysis methodology for 3D data is equivalent to 2-D methodology. The depth error panel is built by extracting the zero offset trace from every downward continuation step of prestack data recorded at the surface. In this paper, the downward operator is a 3-D phase-shift in CMP domain operator.

It is important to point out, that in 3-D the focusing point has associated a depth error and lateral position error in in-line and cross line directions. Therefore, to apply equation (6) and (2), it is necessary to perform a residual migration of depth error gather, then the focusing point, the migrated point and the real point are along the same normal ray.

Figure 3

11/11/1997