# Migration velocity analysis

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

Velocity estimation, CMP stacking, and migration generally are considered independent processes. However, they all have a common theoretical base — the scalar wave equation. Solution of this equation allows downward extrapolation of a seismic wavefield recorded at the earth’s surface. In turn, downward extrapolation provides a basis for CMP stacking and migration [1][2]. Because the processes of CMP stacking and migration require velocity information, they also can be used to obtain a velocity estimate [3] [4].

We now consider the migration process. For a horizontally stratified earth, as in Figure 5.4-1a, we cannot distinguish between a CMP gather and a common-shot gather (CSG). Moreover, since a CSG is a true wavefield created by a single shot and recorded by many receivers, it seems reasonable that the CMP gather in Figure 5.4-1a can be migrated by treating the reflection hyperbola as if it were a diffraction hyperbola. Assuming there is no velocity information, we migrate with various trial constant velocities and evaluate the results. Figure 5.4-2 shows three different attempts at migrating the CMP gather of Figure 5.4-1a. In one attempt (Figure 5.4-2a), too low a velocity was used; hence, the event was undermigrated. In another attempt, too high a velocity was used and the event was overmigrated (Figure 5.4-2c). When the velocity in the migration equals the medium velocity, we expect the diffraction hyperbola to collapse to its apex, which is at the zero-offset trace (Figure 5.4-2b).

What is the implication of this experiment for velocity estimation? Since the correct velocity produces a well-compressed event at the apex of the hyperbola, this velocity can be estimated by evaluating the quality of focusing at zero offset. To evaluate focusing, we pick out the zero-offset traces from the various attempts at migration with different velocities and place them side by side. This produces a display of velocity versus two-way zero-offset time as shown in Figure 5.4-1b.

When Figures 5.4-1b and 5.4-1c are compared, an almost identical character is noted. The resolution of velocity information obtained in the two approaches is equally degraded by data limitations such as maximum source-to-receiver offset and the absence of short-offset traces.

No distinct difference exists between migration and stacking velocity when the subsurface medium is horizontally layered as in Figure 5.4-1. However, for dipping reflectors, the two types of velocity differ. Stacking velocity is sensitive to the dip of the reflecting interface (Section C.3), while, in theory, migration velocity is the medium velocity independent of dip. Therefore, for seismic migration, we must use a velocity field that is corrected for dips present in the data. As a result, any procedure that obtains velocities suitable for migration must use data from a number of neighboring CMP gathers.

Migration velocity analysis can be formulated in much the same way as stacking velocity analysis. To estimate stacking velocities (velocity analysis and statics corrections), in principle, we apply moveout correction to a CMP gather and stack the traces in the gather using a range of constant velocities. Results can be used for velocity determination either in the form of a set of velocity spectra at selected locations along the line or a set of constant-velocity-stack (CVS) panels. To estimate migration velocities, in principle, we migrate prestack data using a range of constant velocities and output selected gathers in their migrated positions or constant-velocity migration (CVM) panels. The key difference between stacking velocity estimation and migration velocity estimation is that the former only requires individual CMP gathers, whereas the latter requires prestack data set in its entirety. The reason for this is that the processes of moveout correction and stacking are confined to within a CMP gather, while the process of migration moves energy spatially from one CMP location to another.

## References

1. Clayton, 1978, Clayton, R., 1978, Common midpoint migration: Stanford Expl. Proj., Rep. No. 14, Stanford University.
2. Yilmaz and Claerbout (1980), Yilmaz, O. and Claerbout, J.F., 1980, Prestack partial migration: Geophysics, 45, 1753–1777.
3. Taner and Koehler, 1969, Taner, M.T. and Koehler, F., 1969, Velocity spectra — digital computer derivation and applications of velocity functions: Geophysics, 32, 859–881.
4. Gardner et al., 1974, Gardner, G.H.F., French, W.S., and Matzuk, T., 1974, Elements of migration and velocity analysis: Geophysics, 39, 811–825.