# Lateral velocity variations

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

Lateral velocity variations often are associated with steep dips. Hence, a depth migration algorithm must not only handle lateral velocity variations but also must image steeply dipping events accurately. The steep-dip implicit and explicit frequency-space migration algorithms described in frequency-space migration in practice are particularly suitable to accommodate lateral velocity variations. In the case of the implicit schemes, the action of the thin-lens term, which accounts for lateral velocity variations, is achieved by a complex multiplication of the wavefield in the frequency-space domain with a velocity-dependent exponential term (Sections D.3 and D.4). In the case of the explicit schemes, lateral velocity variations are accounted for by designing a velocity-dependent, laterally varying extrapolation filter and convolving it with the wavefield in the frequency-space domain (Section D.5).

The problem of lateral velocity variations will be studied using a point diffractor buried in a medium with five different types of velocity-depth models. The first velocity-depth model is shown in Figure 8.0-8. The corresponding zero-offset section consists of an exact diffraction hyperbola. Therefore, only the diffraction term is needed in imaging the scatterer (Section D.3). The surface projection of the point diffractor (the source) coincides with the location of CMP 240, indicated by the vertical arrow, and is aligned with the apex of the diffraction hyperbola. After time migration, the diffraction hyperbola is collapsed to its apex which, in this case, is coincident with the CMP location of the point diffractor.

Consider what happens when the point diffractor is lowered into the second layer as shown in Figure 8.0-9. Raypaths from the scatterer to the surface are bent at the interface between the first and second layers according to Snell’s law of refraction. The zero-offset section is now approximately hyperbolic. From normal moveout, we recall that in a horizontally layered earth model, travel-times are governed by the hyperbolic moveout equation. However, moveout is hyperbolic only within the small-spread approximation. The velocity associated with this approximate hyperbola is the vertical rms velocity down to the diffractor. Suppose the velocity-depth model of Figure 8.0-9 is replaced with that of Figure 8.0-10, where the velocity of the first layer now corresponds to the rms velocity of the diffractor of Figure 8.0-9 (1790 m/s). The zero-offset section associated with this new model is an exact hyperbola (Figure 8.0-10).

The traveltime trajectory in the zero-offset section derived from the original model (Figure 8.0-9) differs negligibly from the hyperbolic traveltime trajectory in the section associated with the replacement model (Figure 8.0-10) only at the far flanks. Therefore, it is reasonable to assume that the zero-offset traveltime trajectory for a point diffractor in a horizontally layered earth model is a hyperbola. The apex of this approximate hyperbola in Figure 8.0-9 coincides with the surface projection of the diffractor, indicated by the arrow. Therefore, only time migration is needed to image a diffractor that is buried in a horizontally layered earth model. This time migration can be performed either by Kirchhoff summation, which uses the rms velocity, or by frequency-space or frequency-wavenumber methods, which honor raypath bendings at the interfaces associated with horizontal layers.

Now consider the point diffractor situated in the third layer as shown in Figure 8.0-11. Now we no longer have even an approximate hyperbolic diffraction response. The traveltime trajectory is skewed so that the apex *A* does not coincide with the lateral position *B* of the diffractor. As expected, time migration partially focuses the energy toward its apex *A*, which is to the left of the actual lateral position of diffractor *B*.

To properly focus the energy and place it laterally to its true subsurface position *B*, depth migration must be performed as shown in Figure 8.0-11. The depth-migrated image is aligned with the true subsurface position *B*. This lateral positioning is accomplished by the thin-lens term. The amount of lateral shift is the lateral distance *AB* between the apex *A* of the skewed diffraction traveltime trajectory and the actual location *B* of the diffractor. This shift depends on the amount of ray bending that occurs at the interfaces above the point diffractor.

**Figure 8.0-8**The response of a point diffractor buried in a constant-velocity medium (top frame) is a hyperbola (middle frame). Time migration (bottom frame) is adequate in imaging this diffractor. The arrows indicate the surface projection of the true lateral position of the diffractor.**Figure 8.0-9**The response of a point diffractor buried in a layered medium (top frame) is approximately a hyperbola (middle frame). Time migration (bottom frame) still is adequate for imaging the diffractor. The arrows indicate the surface projection of the true lateral position of the diffractor.**Figure 8.0-10**The response of a point diffractor buried in a layered medium as in Figure 8.0-9 is nearly equivalent to the response of another point diffractor buried in a constant-velocity medium (top frame), where the constant velocity is the rms value of the layered velocity function at the depth of the original diffractor as in Figure 8.0-9. Traveltimes associated with the zero-offset sections in Figures 8.0-9 and 8.0-10 depart from one another only at the very far flanks of the diffraction curve. Time migration (bottom frame) is adequate for imaging the scatterer.**Figure 8.0-12**The response of a point diffractor buried in a medium with mild to moderate lateral velocity variation (top frame) is a slightly skewed hyperbola with its apex shifted to the left of the true position. Time migration still can be an acceptable way to image the diffractor.

From Figure 8.0-11, note that the apex of the skewed traveltime trajectory *A* coincides with the surface position of the ray that emerges vertically. This special ray is called the *image ray*, and it was first recognized by Hubral ^{[1]}. The image ray associated with the point diffractor in Figure 8.0-11 is roughly at midpoint 200. The diffractor itself is located beneath midpoint 240. Therefore, the lateral shift is equivalent to 40 midpoints.

There is no lateral shift for the horizontally layered earth model (Figure 8.0-9), since there is no lateral velocity variation. The image ray in this case emerges at the surface location of CMP 240 coincident with that of the diffractor. For a mild to moderate lateral velocity variation, as in Figure 8.0-12, the lateral shift is less than 10 midpoints. For some objectives, this small lateral shift and the complete focusing may not be critical; therefore, time migration may be acceptable in lieu of depth migration. In those cases, the coefficient of the thin-lens term is negligibly small, which is why we often can get away with time migration in areas of mild to moderate lateral velocity variations.

The imaging problem is complicated when the overburden is as complex as that in Figure 8.0-13. Here, because of the bowties, a distorted diffraction traveltime trajectory indicates a false structure. The resulting complexity can give rise to more than one image ray. In this case, three image rays emerge at around midpoints 160, 250, and 370. Time migration fails here and imaging of this scatterer can only be achieved by depth migration.

We studied some examples of lateral velocity variations in Figures 8.0-8 through 8.0-13. The image ray behavior and the quality of focusing determine whether time or depth migration should be performed. If the starting and end points of the image ray have the same CMP location (Figure 8.0-9), only time migration is needed. A small amount of lateral deviation of the image ray (Figure 8.0-12) usually implies a well-focused time migration result and hence, a good representation of the geometric form of the subsurface. Large image-ray deviations imply grossly incorrect focusing, thus requiring depth migration rather than time migration (Figure 8.0-11). Finally, if more than one image ray is associated with a subsurface point (Figure 8.0-13), depth migration is imperative.

These observations on the point diffractor models are extended to a velocity-depth model that involves the reflecting interfaces in Figure 8.0-13. The image rays associated with this model are shown in Figure 8.0-14. There is no deviation from the vertical along the image rays down to horizon 2. Therefore, depth migration is not needed to image this horizon. On the other hand, the image rays significantly deviate from the vertical as they travel down to horizons 3 and 4. For example, the image ray starting at CMP location 140 reaches horizon 4 approximately beneath CMP location 180; a lateral shift of 40 midpoints. Proper imaging of these two horizons is achievable only by depth migration.

From Figure 8.0-11, note that time migration collapses the energy to apex *A* of the diffraction curve that coincides with the image ray location at the surface. In principle, the time migration output then can be converted to depth along the image rays, rather than along the vertical rays ^{[1]}. Mapping along the image rays performs some of the action that is associated with the thin-lens term. Remember that at each downward-continuation step, the action of the thin-lens term is equivalent to a time shift that depends on spatial velocity variation. Since the thin-lens and diffraction terms are applied in an alternate manner as the wavefield is downward continued in depth, the effects of these two terms are strongly coupled when the lateral velocity variation is as severe as shown in Figure 8.0-13.

When lateral velocity variation is moderate to strong, these two terms can, in principle, be separated and applied consecutively without significant error ^{[2]}. Full separation implies that a correction for the effects of the thin-lens term can be done either before or after time migration. If the correction is done after time migration, image-ray mapping should be used. If the correction is done before time migration, mapping using vertical time shifts usually is applied. In practice, a correction before time migration often performs better, since it tends to provide a better focused migration result. Nevertheless, all of the above conjectures are related only to now outdated migration algorithms since contemporary implementations of depth migration algorithms are based on splitting, and not separation, of the diffraction and thin-lens terms (Sections D.3 and D.4). It is not a common practice to correct for the effect of the thin-lens term by time-to-depth conversion of time-migrated data using image rays. However, it is a common practice to do time-to-depth conversion of time horizons interpreted from time-migrated data by using image rays (model building).

## References

## See also

- Introduction to earth imaging in depth
- Layer replacement
- 2-D poststack depth migration
- 2-D prestack depth migration
- 3-D poststack depth migration
- 3-D prestack depth migration
- Exercises
- Diffraction and ray theory for wave propagation