# Migration velocities

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

Horizontal displacement during migration is proportional to migration velocity squared (equation 1). Since velocities generally increase with depth, errors in migration are usually larger for deep events than shallow events. Also, the steeper the dip, the more accurate the migration velocities need to be, since displacement is proportional to dip.

 ${\displaystyle d_{x}={\frac {v^{2}t}{4}}{\frac {\Delta t}{\Delta x}},}$ (1)

Figure 4.0-13 shows a portion of a CMP-stacked section after 3-D poststack time migration using a percent range of stacking velocities. Note the subtle under- and overmigration effects on dipping events below the major unconformity represented by the strong, near-horizontal reflection. Events A dipping up to the left and B dipping up to the right cross over one another on sections that correspond to 90 percent and 95 percent of stacking velocities, indicating undermigration. The same events are split away from one another in opposite directions on sections that correspond to 105 percent and 110 percent of stacking velocities. The most overall acceptable image is seen on the section that corresponds to the 100 percent of stacking velocities.

Accuracy in event positioning after migration actually depends on the combined effects of the performance of the migration algorithm used and the velocity errors. For example, the inherently undermigrating character of a 45-degree finite-difference algorithm can be, for an event with a specific dip, coincidentally counterbalanced by the overmigration effect of erroneously too high velocities. In the presence of large vertical velocity gradients, a two-pass 3-D migration can also cause overmigration of steep dips (in the form of lateral translation) even with the correct migration velocities.

Figure 4.0-14a shows a portion of a migrated stacked section. Although this section does not contain steep dips, accurate imaging of the faults along the low-relief structures can be important to the interpreter. Note the slight undermigration, which may be caused by any of the following: (a) error in migration velocities, (b) a dip-limited algorithm that failed to focus the diffraction energy adequately, or (c) a possible 3-D behavior of the geometry of the reflector. The section in Figure 4.0-14a has been migrated with a dip-limited algorithm. Using a proper algorithm, with the same velocities, we get the migrated section in Figure 4.0-14b. The resulting section shows slight overmigration, which can be attributed to errors in migration velocities. Lowering the velocities gives the improved, but not completely accurate, image in Figure 4.0-14c. Perhaps, the remaining issues in imaging may be attributed to 3-D effects.

As demonstrated by the example in Figure 4.0-14, migration results generally are self-evident — under- and overmigration often can be recognized on a migrated section. Problems in imaging often are traced to accuracy in migration velocities. I consider migration velocities as the weak link between the seismic section and the geologic cross-section.

In the next section, basic principles of migration are presented and the Kirchhoff summation, finite-difference, frequency-space, and frequency-wavenumber algorithms are reviewed. Practical aspects of the migration algorithms are expounded in kirchhoff migration in practice through frequency-wavenumber migration in practice. Specifically, key parameters for each category of the migration algorithms are analyzed using appropriate synthetic and field data examples. Further aspects of migration in practice, including spatial aliasing, migration response to random noise, line length, and irregular topography are discussed in further aspects of migration in practice. The problem of conflicting dips with different stacking velocities that requires dip-moveout (DMO) correction and prestack time migration, and the accompanying topic on migration velocity analysis are deferred until dip-moveout correction and prestack migration. The problem of imaging beneath complex structures that requires earth imaging and modeling in depth is discussed in earth imaging in depth and earth modeling in depth, respectively.