User:Ifarley/focusing analysis
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Title  Seismic Data Analysis 

Chapter  Dipmoveout correction and prestack migration 
Contents
Focusing Analysis
The idea of a velocity analysis that is based on differential solutions of the scalar wave equation first was introduced by Doherty and Claerbout).^{[1]} They used the 15degree finitedifference migration algorithm and worked with single CMP gathers. GonzalezSerrano and Claerbout)^{[2]} later extended the wave equation velocity analysis to slantmidpoint coordinates and worked with linearly moveoutcorrected CMP gathers. The method discussed here^{[3]} operates in the Fourier transform domain using the exact form of the doublesquareroot (DSR) operator. Mathematical details of this method are found in Section E.7. Figure 5.423 summarizes the main computational steps involved in this migration velocity analysis based on wavefield extrapolation.
 Starting with the prestack data in midpoint y, offset h, and twoway event time t in the unmigrated position, represented by the wavefield P(y, h, τ = 0, t) at the surface τ = 0, perform 3D Fourier transform. The variable τ is associated with the direction of wave extrapolation, and is related to depth z by τ = 2z/υ, where υ is the medium velocity.
 Specify an extrapolation velocity function that only varies vertically, υ(τ) and apply the extrapolation operator exp(–iωDSRτ/2) to compute the extrapolated wavefield in the transform domain P(k_{y},k_{h},τ,ω) from the surface wavefield in the transform domain P(k_{y},k_{h},τ = 0,ω).
 To obtain the zerooffset image, sum over the offset wavenumber, and thus obtain P(k_{y},h = 0, τ, ω).
 Apply 2D inverse Fourier transform to obtain the zerooffset image P(y, h = 0,τ,t). The image below a midpoint y is contained in the t – τ plane.
 Perform mapping of the variables as described in Section E.6 from τ to υ. The velocity information is given by the envelope of the velocity volume of data P(y, h =0, τ = t, υ).
We now demonstrate the procedure outlined in Figure 5.423 using a synthetic data set. Figure 5.424 shows two commonoffset sections over a number of point scatterers buried in a constantvelocity earth, where υ = 3000 m/s. Using a constant velocity for extrapolation, υ_{e} = 3000 m/s, the t – τ image plane was produced for each midpoint. Two such planes corresponding to midpoints 1 and 5 denoted in Figure 5.424 are shown in Figure 5.425. The υ–τ planes (Figure 5.426) then were generated from the t – τ image planes by the mapping procedure described in Section E.7. Peak amplitudes for all events occur at the correct medium velocity (3000 m/s). We expect the diffraction events in Figure 5.423 to migrate to the apexes beneath midpoint 1, where the point scatterers are located. Note that in Figure 5.425, almost all the energy is in the image plane corresponding to midpoint 1; just five midpoints away, at midpoint 5, the migrated energy is very low.
How do we interpret the t – τ image planes? If we used the true medium velocity in downward extrapolation, then, according to the imaging principle, we would see all the events along the diagonal τ = t, the image line, on the image plane. This happens in Figure 5.425, because a 3000m/s extrapolation velocity was used, which is just the velocity used in generating the model in Figure 5.423. Any displacement of peak energy from the t = τ image line means that the velocity value used for downward extrapolation differs from that of the event. This displacement is also the basis for mapping from the t – τ image plane to the υ – τ plane by equation (E77).
This mapping is investigated further with the modeled data set shown in Figure 5.427, in which velocity increases with depth. In Figure 5.428b, note that the top and middle events fall to the left of the image line suggesting that the velocity used in extrapolation (υ_{e} = 3000 m/s) is greater than the velocities associated with these events. The bottom event falls on the image line, implying that its velocity is nearly the same as the extrapolation velocity. These observations are confirmed in the corresponding υ – τ planes in Figure 5.429. While true stacking velocity values for the three events are 2700, 2850, and 3000 m/s, the velocities interpreted from Figure 5.429b are about 2500, 2800, and 3000 m/s. Thus, the migrationbased velocity estimate for the shallow event is in error by approximately 8 percent.
To determine the reason for the velocity error, we will consider a migrationbased velocity analysis of our synthetic data example that does not involve the approximate mapping step. Figure 5.330a shows a CMP gather from midpoint 1 in the zerodip region of the depthvariable velocity model associated with the constantoffset sections in Figure 5.427. The migration velocity analysis on this gather (Figure 5.430b) was done by extrapolating the surface wavefield P(k_{h}, ω, τ = 0) repeatedly with different constant velocities in steps of Δτ = Δt (the sampling rate). The zerooffset trace from each attempt was collected after this effort, abandoning the rest of the migrated CMP gather.
Interpretation of the velocity analysis in Figure 5.430b reveals correct stacking velocities for the three events, including the shallowest. Clearly the error observed in Figure 5.429 is attributable to the mapping (equation (E100). Note that the error does not occur because of depth variability of velocity, but instead, because the single extrapolation velocity used differed from the medium velocity. The conventional velocity analysis for midpoint 1 of this model data set is shown in Figure 5.430c for comparison. Note the familiar NMO stretching that is apparent in the shallow event. In other respects, both the results (Figures 5.430b and 5.430c) are comparable.
The departure of an event on the t – τ image plane from the t = τ image line is measured by the quantity Δτ as depicted in Figure 5.431a. In some practical implementations, the t – τ image plane is mapped onto the plane of Δτ versus τ as depicted in Figure 5.431c to determine the rms velocity υ(τ) for time migration from the extrapolation velocity υ_{e} (τ). An event with a velocity error υ (τ) – υ_{e} (τ) is represented by an energy maximum either to the left or to the right of the Δτ = 0 line. The δτ(τ) trend can be picked and translated into a velocity trend as depicted in Figure 5.431b. This type of analysis has come to be called focusing analysis in the industry (Faye and Jeannaut, 1986). It has been used in some cases erroneously to estimate and update velocitydepth models used for depth migration. The method can only provide plausable velocity update within the framework of time migration.
Figure 5.432 is a CMP stack from offshore Texas. A 7000ft portion (64 midpoints each with 48 offset traces) of the profile was used for migration velocity analysis. For computational efficiency, the data were windowed into 1024ms time gates with 50 percent overlap. The image planes for one particular midpoint are shown in Figure 5.433. Different extrapolation velocities picked from a specified regional velocity function are used in each time gate. The velocity scan used in mapping is then carried out within a corridor around this function. Because different extrapolation velocities are used in successive segments, a given event appears at different values of τ in adjacent time segments.
The resulting velocity analysis for the central midpoint is shown in Figure 5.434. In conventional practice, to improve the quality of velocity picks, velocity analyses from a number of neighboring CMP gathers often are summed. Figure 5.434c shows the result of stacking velocity analysis for data from the six adjacent CMP gathers indicated in Figure 5.434a. For the migrationbased method, the υ – τ planes corresponding to these gathers were summed. The result is shown in Figure 5.434b. The most obvious difference between the two results is the lack of shallow information in the migrationbased υ – τ plane. This shortcoming is attributed to spatial aliasing and lack of longoffset data in the shallow time gate. The problem can be eliminated partly by increasing the length of the time gate used in the velocity analysis. With the shortcut timewindowing approach described above, the shallowest time segment did not include the largeoffset data necessary for velocity resolution. Because the events have dip, the derived migration velocities are lower (by up to 4.5 percent) than the velocities derived from the stacking velocity analysis.
The velocity analysis described in this section does not handle lateral variations in velocity. It is based on a Fouriertransform domain formulation with only vertically varying velocity used in extrapolation. This method may be particularly efficient for the dipcorrected velocity estimate needed for time migration.
Fowler's velocityindependent prestack migration
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 twoway event time t in the unmigrated position, create a velocity cube  volume of data P(y, υ, τ) in midpoint y, migration velocity υ and twoway 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:
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 (1984) is particularly efficient and elegant. First, we review Fowler's method to create the velocity cube. Refer to the moveout equation (51) and recall that stacking velocity υ_{stk} is dip dependent:
( )
where
( )
Use equation (511) to establish a relationship between the dipdependent stacking velocities υ_{stk} and the dipindependent DMO velocities υ_{DMO} — velocities estimated from dipmoveoutcorrected data:
( )
This equation is the basis for Fowler mapping of υ_{stk} to υ_{DMO}. Note that the process is applied to data in the frequencywavenumber domain. The Fowler mapping is then followed by constantvelocity Stolt mapping (Sections 4.1 and D.7) to map the DMO velocities υ_{DMO} to migration velocities υ_{mig}.
Stolt migration of the dipmoveoutcorrected data volume in the Fourier transform domain P(k_{y},ω_{0}; υ_{DMO}) involves, first, mapping from ω_{0} to ω_{τ} for a specific k_{y} by using the dispersion relation of equation (424b) recast as
( )
where the relationship ω_{τ} = υ_{mig}k_{z}/2 is used. The output of this mapping P(k_{y},ω_{τ}; υ_{mig}) is then scaled by the quantity S given by equation (424c) recast as
( )
Again, the relationship ω_{τ} = υ_{mig}k_{z}/2 is used to obtain equation <xref reftype="dispformula" rid="ch05eq5">(547)</xref> from equation (424c).
Figure 5.435 describes a flowchart for creating the migration velocity volume P(y, υ_{mig}, τ) by Fowler and Stolt mapping.
 Start with data P(y, h, t) in coordinates of midpoint y, offset 2h and event time t in the unmigrated position, and create constantvelocity stack (CVS) volume P(y, υ_{stk},t_{n}) using a range of velocities υ_{stk}, where t_{n} is the event time after constantvelocity normal moveout correction.
 Apply 2D Fourier transform to obtain the CVS cube P(k_{y}, u_{stk},ω) in the frequencywavenumber domain, where k_{y} and ω are the Fourier transform variables associated with the variables y and t_{n}.
 Sort the CVS volume P(k_{y}, υ_{stk},ω) into a set of constantvelocity sections P(k_{y},ω; υ_{stk}).
 Perform the Fowler mapping based on equation <xref reftype="dispformula" rid="ch05eq3">(545)</xref> on each of the velocity sections so as to obtain the DMO velocity volume P(k_{y},ω_{0}; υ_{DMO}).
 Migrate each of the constantvelocity sections of the DMO velocity volume by performing the Stolt mapping based on equations <xref reftype="dispformula" rid="ch05eq4">(546)</xref> and <xref reftype="dispformula" rid="ch05eq5">(547)</xref> so as to obtain the migration velocity volume P(K_{y}, ω_{τ}; υ_{mig}).
 Apply 2D inverse Fourier transform to obtain the migration velocity volume P(y, τ, υ_{mig}).
A variation of Fowler's sequence described above involves creating the CVS cube directly from DMOcorrected data.
 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.
 Create constantvelocity stack (CVS) volume P(y, υ_{DMO},t_{0}) using a range of velocities υ_{DMO}, where t_{0} is the event time after constantvelocity normal moveout correction (Figure 5.436a).
 Sort the CVS volume P(y, υ_{DMO},t_{0}) into a set of cosntantvelocity stacked sections P(y,t_{0}; υ_{DMO}).
 Apply 2D Fourier transform to obtain the CVS cube P(k_{y}, υ_{DMO},ω_{0}) in the frequencywavenumber domain, where k_{y} and ω are the Fourier transform variables associated with the variables y and t_{0}.
 Sort the CVS volume P(k_{y},υ_{DMO},ω_{0}) into a set of constantvelocity sections P(k_{y},ω_{0};υ_{DMO}).
 Migrate each of the constantvelocity sections of the DMO velocity volume by performing the Stolt mapping based on equations <xref reftype="dispformula" rid="ch05eq4">(546)</xref> and <xref reftype="dispformula" rid="ch05eq5">(547)</xref> so as to obtain the migration velocity volume P(y, ω_{τ}, υ_{mig}).
 Apply 2D inverse Fourier transform to obtain the migration velocity volume P(y, τ, υ_{mig}) (Figure 5.436b).
The migration velocity volume P(y, τ, υ_{mig}) shown in Figure 5.436b 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 5.437. The resulting velocity strands shown in Figure 5.438a are interpolated to create the migration velocity field shown in Figure 5.438b. This velocity field then can be used to extract from the volume the section that corresponds to prestack time migration as shown in Figure 5.439. An enlarged view of this section is shown in Figure 5.440. Note the excellent imaging of the steeply dipping fault planes which conflict with the gently dipping strata.
Alternatively, the plane of (υ_{mig},τ) for each midpoint y can be inverse transformed to the plane of (h, τ) associated with the commonreflectionpoint gather derived from prestack time migration. This is then followed by conventional normalmoveout correction and stacking. The resulting section, again, represents the image from prestack time migration.
The data example shown in Figure 5.436 demonstrates the use of the migration velocity volume in deriving a highfidelity image of fault planes from prestack time migration (Figure 5.440). The data example shown in Figure 5.441 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 DMOcorrected CMP gathers were first NMOcorrected using a range of constant velocities and a CVS volume was created. Next, each constantvelocity 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 5.441. Note the vertical variation in velocities on the endon 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 5.442a. 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 5.442a), an rms velocity surface is generated. The picked velocity strands are shown in Figure 5.442b embedded within the migration velocity volume, and the rms velocity field is shown in Figure 5.443a as a colorcoded surface extracted from within the migration velocity volume. A quality control of the rms velocity surface can be made by intersecting it with the crosssections of the migration velocity volume at selected CMP locations as shown in Figure 5.443b.
The prestack timemigrated section is a byproduct 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 5.444a. The conventional 2D display of the prestack timemigrated section is then created by simply collapsing the sculpted image surface onto a 2D plane (Figure 5.444b). A closeup view of the prestack timemigrated section shows accurate imaging of the steeply dipping event (Figure 5.445).
Figure 5.444 (a) The image surface associated with prestack time migration that was extracted from the migration velocity volume shown in Figure 5.441a by sculpting the amplitudes over the rms velocity surface shown in Figure 5.443a, (b) the image surface as in (a) after collapsing it onto a 2D section associated with prestack time migration.
Exercises
 Exercise 1
 Consider the application of DMO correction to data referred to a floating datum represented by a smooth form of an irregular topographic surface and data referenced to a flat datum below. Which DMO correction would have more effect on the data?
 Exercise 2
 Refer to Fowler's velocityindependent prestack migration technique for migration velocity analysis described in Section 5.4. Suppose you have transformed the prestack data from offset to velocity space using 30 constant velocity values from 2000 m/s to 4900 m/s using an increment of 100 m/s. Can you create additional constantvelocity panels such that the increment is 50 m/s by poststack migration rather than prestack migration or trace interpolation? If so, what velocity would you use for poststack migration to create the panel for 3050m/s velocity.
 Exercise 3
 Derive Bancroft's equivalent offset equation (E72b) from the nonzerooffset traveltime equation (E67).
 Exercise 4
 Suppose you have two events with conflicting dips of the same magnitude, but in opposite directions, associated with a reflector within a fault block and the fault plane itself. Can these two events be distinguished on a velocity spectrum computed from a CMP gather before DMO correction at a location just above the surface point where the two events intersect one another?
See also
References
 ↑ Doherty, S., M. and Claerbout, J., F., 1974, Velocity analysis based on the wave equation: Stanford Expl. Proj., Rep. No. 1, Stanford University.
 ↑ GonzalezSerrano, A and Claerbout, J., F., 1979, Waveequation velocity analysis: Stanford Expl. Proj., Rep. No. 16, Stanford University.
 ↑ Yilmaz, O. and Chambers, R., 1984, Migration velocity analysis by wave field extrapolation: Geophysics, 49, 1664–1674.
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