The idea of a velocity analysis that is based on differential solutions of the scalar wave equation first was introduced by Doherty and Claerbout). They used the 15-degree finite-difference migration algorithm and worked with single CMP gathers. Gonzalez-Serrano and Claerbout) later extended the wave equation velocity analysis to slant-midpoint coordinates and worked with linearly moveout-corrected CMP gathers. The method discussed here operates in the Fourier transform domain using the exact form of the double-square-root (DSR) operator. Mathematical details of this method are found in Section E.7. Figure 5.4-23 summarizes the main computational steps involved in this migration velocity analysis based on wavefield extrapolation.
- Starting with the prestack data in midpoint y, off-set h, and two-way event time t in the unmigrated position, represented by the wavefield P(y, h, τ = 0, t) at the surface τ = 0, perform 3-D Fourier transform. The variable τ is associated with the direction of wave extrapolation, and is related to depth z by τ = 2z/u, where u is the medium velocity. * Specify an extrapolation velocity function that only varies vertically, u(τ) and apply the extrapolation operator exp(—iωDSRτ/2) to compute the extrapolated wavefield in the transform domain P(ky,kh,τ,ω) from the surface wavefield in the transform domain P(ky,kh,τ = 0,ω). * To obtain the zero-offset image, sum over the offset wavenumber, and thus obtain P(ky,h = 0,τ = 0, τ, ω). * Apply 2-D inverse Fourier transform to obtain the zero-offset 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 u. The velocity information is given by the envelope of the velocity volume of data P(y, h =0,r = t, u).
We now demonstrate the procedure outlined in Figure 5.4-23 using a synthetic data set. Figure 5.4-24 shows two common-offset sections over a number of point scatterers buried in a constant-velocity earth, where u = 3000 m/s. Using a constant velocity for extrapolation, ue = 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 5.4-24 are shown in Figure 5.4-25. The u—τ planes (Figure 5.4-26) 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 5.4-23 to migrate to the apexes beneath midpoint 1, where the point scatterers are located. Note that in Figure 5.4-25, 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 5.4-25, because a 3000-m/s extrapolation velocity was used, which is just the velocity used in generating the model in Figure 5.4-23. 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 u — τ plane by equation (E-77).
This mapping is investigated further with the modeled data set shown in Figure 5.4-27, in which velocity increases with depth. In Figure 5.4-28b, note that the top and middle events fall to the left of the image line suggesting that the velocity used in extrapolation (u 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 u — τ planes in Figure 5.4-29. While true stacking velocity values for the three events are 2700, 2850, and 3000 m/s, the velocities interpreted from Figure 5.4-29b are about 2500, 2800, and 3000 m/s. Thus, the migration-based 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 migration-based velocity analysis of our synthetic data example that does not involve the approximate mapping step. Figure 5.3-30a shows a CMP gather from midpoint 1 in the zero-dip region of the depth-variable velocity model associated with the constant-offset sections in Figure 5.4-27. The migration velocity analysis on this gather (Figure 5.4-30b) was done by extrapolating the surface wavefield P(kh , ω, τ = 0) repeatedly with different constant velocities in steps of Δτ = Δt (the sampling rate). The zero-offset trace from each attempt was collected after this effort, abandoning the rest of the migrated CMP gather.
Interpretation of the velocity analysis in Figure 5.4-30b reveals correct stacking velocities for the three events, including the shallowest. Clearly the error observed in Figure 5.4-29 is attributable to the mapping (equation (E-100). 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 5.4-30c for comparison. Note the familiar NMO stretching that is apparent in the shallow event. In other respects, both the results (Figures 5.4-30b and 5.4-30c) 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 5.4-31a. In some practical implementations, the t — τ image plane is mapped onto the plane of Δτ versus τ as depicted in Figure 5.4-31c to determine the rms velocity u(τ) for time migration from the extrapolation velocity u e (τ). An event with a velocity error u (τ) — u 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 5.4-31b. 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 velocity-depth models used for depth migration. The method can only provide plausable velocity update within the framework of time migration.
Figure 5.4-32 is a CMP stack from offshore Texas. A 7000-ft 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 1024-ms time gates with 50 percent overlap. The image planes for one particular midpoint are shown in Figure 5.4-33. 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 5.4-34. In conventional practice, to improve the quality of velocity picks, velocity analyses from a number of neighboring CMP gathers often are summed. Figure 5.4-34c shows the result of stacking velocity analysis for data from the six adjacent CMP gathers indicated in Figure 5.4-34a. For the migration-based method, the u — τ planes corresponding to these gathers were summed. The result is shown in Figure 5.4-34b. The most obvious difference between the two results is the lack of shallow information in the migration-based u — τ plane. This shortcoming is attributed to spatial aliasing and lack of long-offset 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 time-windowing approach described above, the shallowest time segment did not include the large-offset 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 Fourier-transform domain formulation with only vertically varying velocity used in extrapolation. This method may be particularly efficient for the dip-corrected velocity estimate needed for time migration.
- Doherty, S., M., Claerbout, J., F., 1974, Velocity analysis based on the wave equation: Stanford Expl. Proj., Rep. No. 1, Stanford University.
- Gonzalez-Serrano, AClaerbout, J., F., 1979, Wave-equation velocity analysis: Stanford Expl. Proj., Rep. No. 16, Stanford University.
- Yilmaz, O., Chambers, R., 1984, Migration velocity analysis by wave field extrapolation: Geophysics, 49, 1664–1674.