Fowler’s velocity-independent prestack migration

Seismic Data Analysis

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

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.

Figure 5.4-35  A flowchart of an algorithm to create a migration velocity cube.

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 v_stk 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 v_stk and the dip-independent DMO velocities v_DMO — 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 v_stk to v_DMO. 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 v_DMO to migration velocities v_mig.

Stolt migration of the dip-moveout-corrected data volume in the Fourier transform domain P(k_y, ω₀; v_DMO) involves, first, mapping from ω₀ to ω_τ for a specific k_y 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 ω_τ = v_migk_z/2 is used. The output of this mapping P(k_y, ω_τ; v_mig) 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 ω_τ = v_migk_z/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, v_mig, 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, v_stk, t_n) using a range of velocities v_stk, where t_n is the event time after constant-velocity normal moveout correction.
2. Apply 2-D Fourier transform to obtain the CVS cube P(k_y, v_stk, ω) in the frequency-wavenumber domain, where k_y and ω are the Fourier transform variables associated with the variables y and t_n.
3. Sort the CVS volume P(k_y, v_stk, ω) into a set of constant-velocity sections P(k_y, ω; v_stk).
4. Perform the Fowler mapping based on equation (45) on each of the velocity sections so as to obtain the DMO velocity volume P(k_y, ω₀; v_DMO).
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(k_y, ω_τ; v_mig).
6. Apply 2-D inverse Fourier transform to obtain the migration velocity volume P(y, τ, v_mig).

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, v_DMO, t₀) using a range of velocities v_DMO, where t₀ is the event time after constant-velocity normal moveout correction (Figure 5.4-36a).
3. Sort the CVS volume P(y, v_DMO, t₀) into a set of constant-velocity stacked sections P(y, t₀; v_DMO).
4. Apply 2-D Fourier transform to obtain the CVS cube P(k_y, v_DMO, ω₀) in the frequency-wavenumber domain, where k_y and ω are the Fourier transform variables associated with the variables y and t₀.
5. Sort the CVS volume P(k_y, v_DMO, ω₀) into a set of constant-velocity sections P(k_y, ω₀; v_DMO).
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(k_y, ω_τ; v_mig).
7. Apply 2-D inverse Fourier transform to obtain the migration velocity volume P(y, τ, v_mig) (Figure 5.4-36b).

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  Figure 5.4-36  (a) A velocity cube computed from DMO-corrected gathers; (b) the same data as in (a) after Stolt migration of each of the constant-velocity panels.

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  Figure 5.4-37  (a) A time slice from the migration velocity cube shown in Figure 5.4-36b, (b) the same time slice as in (a) with velocity picking. Red and dark blue curves correspond to primaries and multiples, respectively.

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  Figure 5.4-38  (a) Velocity strands picked from the time slices of the migration velocity cube shown in Figure 5.4-36b as demonstrated in Figure 5.4-37b; (b) the color-coded velocity field computed by interpolation of the velocity strands shown in (a).

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  Figure 5.4-39  (a) The prestack time-migrated section extracted from the migration velocity cube of Figure 5.4-36b coincident with the surface that corresponds to the velocity field in Figure 5.4-38b, (b) the same section as in (a) with the viewing axis perpendicular to the plane of the paper.

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  Figure 5.4-40  The prestack time-migrated section as in Figure 5.4-39b collapsed onto a 2-D section.

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  Figure 5.4-41  (a) A migration velocity volume, (b) the same volume viewed end-on showing the vertical variation in rms velocity at a given CMP location.

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  Figure 5.4-42  (a) A time slice from the migration velocity volume shown in Figure 5.4-41a with the interpretation of the rms velocity as a function of midpoint location, (b) the migration velocity volume as in Figure 5.4-41a with the picked velocity strands as in (a).

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  Figure 5.4-43  (a) The color-coded optimum rms velocity surface extracted from the migration velocity volume shown in Figure 5.4-41a, (b) the same view as in Figure 5.4-41b with the intersection of the rms velocity surface showing the vertical velocity variation at a given CMP location.

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  Figure 5.4-44  (a) The image surface associated with prestack time migration that was extracted from the migration velocity volume shown in Figure 5.4-41a by sculpting the amplitudes over the rms velocity surface shown in Figure 5.4-43a, (b) the image surface as in (a) after collapsing it onto a 2-D section associated with prestack time migration.

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  Figure 5.4-45  A close-up view of the prestack time-migrated section as in Figure 5.4-44b.

The migration velocity volume P(y, t, v_mig) 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 (v_mig, 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).

See also

• Prestack Stolt migration
• Common-offset migration of DMO-corrected data
• Prestack Kirchhoff migration
• Velocity analysis using common-reflection-point gathers
• Focusing analysis

External links

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