Page 518 - Acquisition and Processing of Marine Seismic Data

11.3 FINITE-DIFFERENCE MIGRATION 509

FIG. 11.17 The effect of migration velocity errors on the collapse of a diffraction hyperbola in a constant velocity medium
by poststack Kirchhoff time migration. Velocity of the zero-offset section is 1500 m/s. Over- and undermigration occur when
the migration velocity is faster or slower than the actual velocity of the medium, respectively.

diffraction hyperbolas shorten and arrive earlier 11.3.1 Depth Step Size
(Fig. 11.19). When the receivers arrive at the apex,
the hyperbola is completely collapsed to a single Computation of the wave field towards the
point at t ¼ 0ms (Fig. 11.19C). An undermigra- depth is achieved by discrete depth steps,
tion occurs and the hyperbola cannot be col- which determine the performance and effec-
lapsed perfectly if the downward continuation tiveness of the finite-difference migration algo-
is completed before the source depth is reached rithm. Fig. 11.20 shows the effect of depth step
(Fig.11.19B).Thesamesituationalsooccurswhen size on the collapse of a theoretical diffraction
a slower velocity for the migration is used. hyperbola. The minimum depth step size is
Inversely, if the downward continuation is con- the sampling rate of the input. Higher depth
tinued after the source depth is reached, then a step sizes result in undermigration and the dif-
mirror image of the diffraction hyperbola is fraction hyperbolas cannot be collapsed. Yılmaz
obtained and an overmigration occurs (2001) suggests that this situation also results in
(Fig. 11.19D and E), such as occurs in the case kinks and a specific noise called dispersion
when a faster velocity for the migration is used. along the flanks of the diffraction hyperbola.