# Stolt stretch factor

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 |

As discussed in migration principles, the generalized Stolt method of migration involves converting the time section to an approximately constant-velocity section, which then is migrated by the constant-velocity Stolt algorithm. This conversion is essentially stretching in the vertical (time) direction. Once the section is migrated in the stretched domain, it is converted back to the original time domain. The generalized Stolt method must be distinguished from the constant-velocity algorithm. The constant-velocity algorithm is accurate for dips up to 90 degrees for a constant-velocity medium. The generalized method approximately accounts for velocity variations by stretching the section.

Stretching is defined by the stretch factor *W*. In his original paper, Stolt ^{[1]} discusses implementation of the *W* factor. Although *W* is a complicated function of velocity and stretch coordinate variables, it often is set to a scalar (Section D.7). The theoretical range of *W* is between 0 and 2.

To understand the effects of the stretch factor, refer to the impulse responses in Figure 4.5-16, where a single, isolated wavelet on a single trace is migrated using various stretch factors. Here, *W* =1 corresponds to the exact constant-velocity Stolt algorithm. So, setting *W* < 1 compresses the impulse response inward along its steep flanks, while setting *W* > 1 opens it up. Thus, the value of *W* partially controls the aperture of the generalized Stolt algorithm. The farther *W* is from 1, the more limited the aperture becomes. A value of *W* < 1 implies undermigration at steeper dips, while a value of *W* > 1 implies overmigration at steeper dips, if the medium velocity is constant.

Although not strictly implied by the impulse responses in Figure 4.5-16, when using a stretch factor different from 1, the Stolt algorithm tries to emulate a wavefront in a variable velocity medium ^{[1]}, while compromising on the ability to migrate steeper dips. Experience has proven that the Stolt migration with stretch produces acceptable results provided velocity variations are within limits of time migration.

Consider the zero-offset section and the migration results in Figure 4.5-17. Stretch factor *W* = 1 produces the best migrated section because the zero-offset section was modeled using a constant-velocity value. For 0 < *W* < 1, the algorithm produces an undermigrated section, while for 1 < *W* < 2, it produces an overmigrated section. These observations are in agreement with the impulse responses in Figure 4.5-16. The near-vertical streaks in the section with *W* = 1.95 represent wraparound artifacts.

The generalized Stolt algorithm produces the best result when *W* =1, provided the medium velocity is constant. Since this is never the case, we should examine the algorithm for a vertically varying velocity medium. Figure 4.5-18 shows the impulse responses for different values of *W*. Velocity varies linearly from *t* = 0 to *t* = 4 s between 2000 and 4000 m/s. For different *W* values, the portions of the wavefronts that best match the desired migration using the phase-shift method are between the solid lines. For a vertically varying velocity medium, *W* = 1 is no longer the desired factor. In Figure 4.5-18, accuracy over the widest range of dip angles with the Stolt method is attained when *W* = 0.6. In general, the larger the velocity gradient, the farther the optimum *W* is from 1. Strictly speaking, the optimum value for *W* is even different at different times.

In practice, wavefront plots, like those in Figure 4.5-18, can be generated using both the phase-shift and Stolt methods for a vertically varying regional velocity function. The *W* factor that yields the best fit at the largest angular aperture is used then to migrate the data with the Stolt method.

**Figure 4.5-16**Tests for the stretch factor in Stolt migration: By varying the stretch factor*W*, the impulse response of the exact 90-degree migration operator (semicircle) is modified. For comparison, the desired response has been superimposed on the Stolt migration impulse responses.**Figure 4.5-17**Tests for the stretch factor in Stolt migration:*W*< 1 causes undermigration, and*W*> 1 causes overmigration. (Modeling courtesy Union Oil Company.)**Figure 4.5-18**Tests for the stretch factor in Stolt migration: The medium velocity varies vertically from 2000 m/s at*t*= 0 to 4000 m/s at*t*= 4 s.

To circumvent the difficulty of defining an optimum stretch factor *W*, Beasley and Lynn ^{[2]} suggested applying the constant-velocity Stolt migration in a cascaded manner. The idea is based on a clever representation of a vertically varying velocity function by a set of constant velocities, which are then used to perform cascaded migration (finite-difference migration in practice). Since each migration stage is done by using a constant velocity, the stretch factor *W* is by default set to 1. Of course, representation of a vertically varying velocity function by a set of constant velocities is only an approximation that can be valid for small vertical gradients.

Figures 4.5-19 and 4.5-20 show Stolt migrations of the CMP stack in Figure 4.3-2a using different values of the *W* factor. Migration velocities are varied only in the vertical direction. Figure 4.5-21 is a sketch of the migration results for the diffraction *D* off the tip of the salt diapir and steeply dipping event *B* off the flank of the salt diapir. The best match between the desired migration and the Stolt method with stretch is for *W* = 0.5.

**Figure 4.5-19**Tests for the stretch factor*W*in Stolt migration: (a) Desired migration, (b)*W*= 0.001, and (c)*W*= 0.5. The input CMP stack is shown in Figure 4.3-2a.**Figure 4.5-20**Tests for the stretch factor*W*in Stolt migration: (a)*W*= 0.7, (b)*W*= 1, and (c)*W*= 1.5. The input CMP stack is shown in Figure 4.3-2a and the desired migration is shown in Figure 4.5-19a.**Figure 4.3-2**(a) CMP stack, (b) desired migration by phase-shift method, (c) 15-degree finite-difference migration. The finite-difference migration based on the parabolic equation has the inherent property of undermigrating the steep flank of the diffraction and the steeply dipping event. See Figure 4.3-3 for a sketch of the migration results.**Figure 4.5-21**The combined results of the migrations from Figures 4.5-19 and 4.5-20.*B*= dipping event before and*A*= dipping event after desired migration,*D*= diffraction before and*D′*= diffraction after Stolt migration with stretch. Numbers represent different stretch factors*W*.

## References

- ↑
^{1.0}^{1.1}Stolt (1978), Stolt, R.H., 1978, Migration by Fourier transform: Geophysics, 43, 23–48. - ↑ Beasley and Lynn (1992), Beasley, C., Lynn, W. S., Larner, K. L. and Nguyen, H. V., 1992, Extended Stolt
*f − k*migration: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 551–553.