Velocity analysis of PS data
From the raypath geometry of Figure 11.6-32, it follows that
where t is the two-way PS reflection traveltime from the source to the conversion point to the receiver, and z is the reflector depth.
Set x = xP = 0 in equation (74) to see that the two-way zero-offset PS traveltime is given by
where γ = α/β.
Equation (76) describes the PS-wave moveout observed on a CCP gather. Although it is derived for a single flat layer in a constant-velocity medium, this equation also is applicable to a horizontally layered earth model. In that case, α and β refer to the P- and S-wave rms velocities.
Note from equation (76) that the asymmetric raypath associated with the PS reflection shown in Figure 11.6-32 gives rise to a nonhyperbolic moveout even in the case of a flat reflector in a constant-velocity medium. A way to avoid dealing with nonhyperbolic moveout would be to make the small-spread approximation and consider the best-fit hyperbola
to the traveltime trajectory associated with a PS reflection on a CCP gather . In equation (77), t and t0 mean the same traveltimes as in equation (76), and vNMO is the moveout velocity for PS-wave derived from the best-fit hyperbola; as such, it is neither the P-wave velocity α nor the S-wave velocity β. In fact, the Taylor expansion of equation (74) yields the relation . By assuming a hyperbolic moveout described by equation (77), the PS data can be corrected for moveout and stacked in the same manner as for the PP data.
Figure 11.6-32 Geometry of a common-conversion-point (CCP) raypath used to derive the reflection traveltime equation (74) for the PS-wave.
Practical experience, however, points to the unavoidable fact that the PS data exhibit strong nonhyperbolic moveout behavior. Shown in Figure 11.6-33 is a CCP gather after hyperbolic moveout correction using equation (77) and nonhyperbolic moveout correction using equation (76). Note the overcorrection at far offsets within 0-2.5 s.
The need for nonhyperbolic moveout correction for the PS data makes it compelling to conduct a multiparameter velocity analysis. Unlike velocity analysis using the hyperbolic moveout equation (77), where we only need to scan for one parameter, vNMO, velocity analysis using the nonhyperbolic moveout equation (76) suggests scanning for three parameters — the PP-wave velocity α, the velocity ratio γ = α/β, and the CCP displacement xP. But, in practice, we do not have to scan for all three parameters. Instead, the iterative procedure described below may be followed.
- To begin with, note that the PP-wave velocity α can be estimated directly by velocity analysis of the PP data set itself.
- We may assume an initial value for the velocity ratio γ = α/β and estimate an initial value for xP using equation (73a).
- Knowing α and xP, use the nonhyperbolic moveout equation (76) to scan for γ as a function of t0. Figure 11.6-34 shows a γ-spectrum computed from the CCP gather in Figure 11.6-33a.
- Pick a function γ(t0) at each CCP analysis location along the line over the survey area and derive a γ(x, t0)-section as shown in Figure 11.6-35.
- Use the γ(x, t0)-section and the PP-wave velocity α to calculate an updated value for xP(t0) from equation (72c).
- Substitute the updated xP(t0) and the estimated γ and α into equation (76) to perform the nonhyperbolic moveout correction (Figure 11.6-33c).
Another strategy for velocity analysis of the PS-wave data is the direct estimation of the PS-wave velocity β, rather than estimating the velocity ratio γ. Return to equation (76) and rewrite it explicitly in terms of α and β as
Figure 11.6-34 Analysis of CCP gather in Figure 11.6-33a for the velocity ratio, γ = α/β. Figure courtesy .
Figure 11.6-35 The velocity-ratio section derived from the analysis as in Figure 11.6-34. Figure courtesy .
- Again, estimate the PP-wave velocity α as before using the PP data set itself.
- Also, assume an initial value for γ = α/β and estimate an initial value for xP using equation (73a).
- Knowing α and xP, use the nonhyperbolic moveout equation (78) to scan for β as a function of t0. Figure 11.6-36 shows a CCP gather and the computed β-spectrum. Compare with the α-spectrum in Figure 11.6-20 and note the difference in the velocity ranges in the two spectra.
- Pick a function β(t0) at each CCP analysis location along the line over the survey area and derive a β(x, t0)-section.
- Use the β(x, t0)-section and the PP-wave velocity α to calculate an updated value for xP(t0) from equation (72c).
- Substitute the updated xP(t0) and the estimated β and α into equation (78) to perform the non-hyperbolic moveout correction.
Figure 11.6-37 shows the PS-wave stack based on the alternative procedure described above. Note the differences between this section and the PP-wave stack shown in Figure 11.6-21. Specifically, the PS-wave stack shows some interesting reflector geometry between 4-5 s; this behavior is absent within the equivalent time window (2-2.5 s) in the PP-wave stack.
One subtle issue is related to the time at which a is specified in equations (76) and (78). The two-way zero-offset time t0 in these equations is associated with the PS-wave; whereas, α is estimated at two-way zero-offset time associated with the PP-wave. To distinguish the two zero-offset times, first, rewrite equation (75) for the PS two-way time
For the same depth z, the PP two-way time is given by
Now eliminate z between the two equations to get the relation between the PP and PS zero-offset times
or in terms of γ = α/β
- Tessmer and Behle, 1988, Tessmer, G. and Behle, A., 1988, Common reflection point data stacking technique for converted waves: Geophys. Prosp., 36, 671–688.
- Fromm et al., 1985, Fromm, G., Krey, T., and Wiest, B., 1985, Static and dynamic corrections, in Dohr, G., Ed., Seismic Shear Waves: Handbook of Geophysical Exploration, vol. 15a: Geophysical Press, 191–225.
- Li and Yuan, 1999, Li, Xiang-Yang and Yuan, Jianxin, 1999, Developing and exploiting effective techniques to overcome the difficulties of 4-C seismic reservoir monitoring in deep water: Edinburgh Anisotropy Project Ann. Report, 129–155.
- 4-C seismic method
- Recording of 4-C seismic data
- Gaiser’s coupling analysis of geophone data
- Processing of PP data
- Rotation of horizontal geophone components
- Common-conversion-point binning
- Dip-moveout correction of PS data
- Migration of PS data