# Fowler’s velocity-independent prestack migration

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

We now restate the underlying principle for migration velocity analysis:

Starting with the prestack volume of data P(y, h, t) in midpoint y, offset h and two-way event time t in the unmigrated position, create a velocity cube — volume of data P(y, v, τ) in midpoint y, migration velocity v and two-way event time τ in the migrated position. Within the context of time migration, the output time variable τ is related to depth by way of vertical stretch: τ = 2z/v.

Although the velocity cube can be created by means of some of the migration velocity analysis techniques described in this section, a variation of the method by Fowler  is particularly efficient and elegant. First, we review Fowler’s method to create the velocity cube. Refer to the moveout equation (1) and recall that stacking velocity vstk is dip dependent:

 $t^{2}=t_{0}^{2}+{\frac {4h^{2}{\rm {cos}}^{2}\phi }{v^{2}}},$ (1)

 $t^{2}=t_{0}^{2}+{\frac {4h^{2}}{v_{stk}^{2}}},$ (43)

where

 $v_{stk}={\frac {v_{DMO}}{\cos \phi }}.$ (44)

Use equation (11) to establish a relationship between the dip-dependent stacking velocities vstk and the dip-independent DMO velocities vDMO — velocities estimated from dip-moveout-corrected data:

 ${\rm {sin}}\phi ={\frac {vk_{y}}{2{\omega _{0}}}},$ (11)

 $v_{stk}={\frac {v_{DMO}}{\sqrt {1-\left({\frac {v_{DMO}k_{y}}{2\omega _{0}}}\right)^{2}}}}.$ (45)

This equation is the basis for Fowler mapping of vstk to vDMO. Note that the process is applied to data in the frequency-wavenumber domain. The Fowler mapping is then followed by constant-velocity Stolt mapping (migration principles and D.7) to map the DMO velocities vDMO to migration velocities vmig.

Stolt migration of the dip-moveout-corrected data volume in the Fourier transform domain P(ky, ω0; vDMO) involves, first, mapping from ω0 to ωτ for a specific ky by using the dispersion relation of equation (4-24b) recast as

 $\omega _{0}={\sqrt {\omega _{\tau }^{2}+{\frac {v_{mig}^{2}k_{y}^{2}}{4}}}},$ (46)

where the relationship ωτ = vmigkz/2 is used. The output of this mapping P(ky, ωτ; vmig) is then scaled by the quantity S given by equation (4-24c) recast as

 $S={\frac {v_{mig}}{2}}{\frac {\omega _{\tau }}{\sqrt {\omega _{\tau }^{2}+{\frac {v_{mig}^{2}k_{y}^{2}}{4}}}}}.$ (47)

Again, the relationship ωτ = vmigkz/2 is used to obtain equation (47) from equation (4-24c).

Figure 5.4-35 describes a flowchart for creating the migration velocity volume P(y, vmig, τ) by Fowler and Stolt mapping.

1. Start with data P(y, h, t) in coordinates of midpoint y, offset 2h and event time t in the unmigrated position, and create constant-velocity stack (CVS) volume P(y, vstk, tn) using a range of velocities vstk, where tn is the event time after constant-velocity normal moveout correction.
2. Apply 2-D Fourier transform to obtain the CVS cube P(ky, vstk, ω) in the frequency-wavenumber domain, where ky and ω are the Fourier transform variables associated with the variables y and tn.
3. Sort the CVS volume P(ky, vstk, ω) into a set of constant-velocity sections P(ky, ω; vstk).
4. Perform the Fowler mapping based on equation (45) on each of the velocity sections so as to obtain the DMO velocity volume P(ky, ω0; vDMO).
5. Migrate each of the constant-velocity sections of the DMO velocity volume by performing the Stolt mapping based on equations (46) and (47) so as to obtain the migration velocity volume P(ky, ωτ; vmig).
6. Apply 2-D inverse Fourier transform to obtain the migration velocity volume P(y, τ, vmig).

A variation of Fowler’s sequence described above involves creating the CVS cube directly from DMO-corrected data.

1. Start with data P(y, h, t) in coordinates of midpoint y, offset 2h and event time t in the unmigrated position, and apply DMO correction followed by inverse NMO correction.
2. Create constant-velocity stack (CVS) volume P(y, vDMO, t0) using a range of velocities vDMO, where t0 is the event time after constant-velocity normal moveout correction (Figure 5.4-36a).
3. Sort the CVS volume P(y, vDMO, t0) into a set of constant-velocity stacked sections P(y, t0; vDMO).
4. Apply 2-D Fourier transform to obtain the CVS cube P(ky, vDMO, ω0) in the frequency-wavenumber domain, where ky and ω are the Fourier transform variables associated with the variables y and t0.
5. Sort the CVS volume P(ky, vDMO, ω0) into a set of constant-velocity sections P(ky, ω0; vDMO).
6. Migrate each of the constant-velocity sections of the DMO velocity volume by performing the Stolt mapping based on equations (46) and (47) so as to obtain the migration velocity volume P(ky, ωτ; vmig).
7. Apply 2-D inverse Fourier transform to obtain the migration velocity volume P(y, τ, vmig) (Figure 5.4-36b).

The migration velocity volume P(y, t, vmig) shown in Figure 5.4-36b can be visualized and interpreted to derive a migration velocity field. For spatial consistency, velocity picking should be done on time slices from the migration velocity volume as shown in Figure 5.4-37. The resulting velocity strands shown in Figure 5.4-38a are interpolated to create the migration velocity field shown in Figure 5.4-38b. This velocity field then can be used to extract from the volume the section that corresponds to prestack time migration as shown in Figure 5.4-39. An enlarged view of this section is shown in Figure 5.4-40. Note the excellent imaging of the steeply dipping fault planes which conflict with the gently dipping strata.

Alternatively, the plane of (vmig, τ) for each midpoint y can be inverse transformed to the plane of (h, τ) associated with the common-reflection-point gather derived from prestack time migration. This is then followed by conventional normal-moveout correction and stacking. The resulting section, again, represents the image from prestack time migration.

The data example shown in Figure 5.4-36 demonstrates the use of the migration velocity volume in deriving a high-fidelity image of fault planes from prestack time migration (Figure 5.4-40). The data example shown in Figure 5.4-41 demonstrates the use of the migration velocity volume in imaging steeply dipping event. The migration velocity volume was created using the procedure described above. Specifically, the DMO-corrected CMP gathers were first NMO-corrected using a range of constant velocities and a CVS volume was created. Next, each constant-velocity stacked section was migrated using the Stolt method and the constant velocity associated with that section. The resulting migration velocity volume is shown in Figure 5.4-41. Note the vertical variation in velocities on the end-on view of the volume that represents the plane of velocity versus time.

The migration velocity volume is interpreted by picking the primary velocity trend from selected time slices as shown in Figure 5.4-42a. Note the lateral variation in velocities which is captured by continuous picking along the midpoint axis. By interpolating the velocity strands resulting from the interpretation of selected time slices (Figure 5.4-42a), an rms velocity surface is generated. The picked velocity strands are shown in Figure 5.4-42b embedded within the migration velocity volume, and the rms velocity field is shown in Figure 5.4-43a as a color-coded surface extracted from within the migration velocity volume. A quality control of the rms velocity surface can be made by intersecting it with the cross-sections of the migration velocity volume at selected CMP locations as shown in Figure 5.4-43b.

The prestack time-migrated section is a by-product of the migration velocity analysis described here. Specifically, the image surface associated with the prestack time migration is obtained by sculpting the amplitudes from within the migration velocity volume over the rms velocity surface as shown in Figure 5.4-44a. The conventional 2-D display of the prestack time-migrated section is then created by simply collapsing the sculpted image surface onto a 2-D plane (Figure 5.4-44b). A close-up view of the prestack time-migrated section shows accurate imaging of the steeply dipping event (Figure 5.4-45).