# Velocity analysis of PS data

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

From the raypath geometry of Figure 11.6-32, it follows that

 ${\displaystyle t={\frac {1}{\alpha }}{\sqrt {x_{P}^{2}+z^{2}}}+{\frac {1}{\beta }}{\sqrt {(x-x_{P})^{2}+z^{2}}},}$ (74)

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

 ${\displaystyle t_{0}=\left({\frac {1}{\alpha }}+{\frac {1}{\beta }}\right)z.}$ (75)

Substitute equation (75) into equation (74) for the depth variable z to get

 ${\displaystyle t={\frac {1}{\alpha }}{\sqrt {x_{P}^{2}+{\frac {\alpha ^{2}}{(\gamma +1)^{2}}}t_{0}^{2}}}+{\frac {\gamma }{\alpha }}{\sqrt {(x-x_{P})^{2}+{\frac {\alpha ^{2}}{(\gamma +1)^{2}}}t_{0}^{2}}},}$ (76)

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

 ${\displaystyle t=t_{0}^{2}+{\frac {x^{2}}{v_{NMO}^{2}}}}$ (77)

to the traveltime trajectory associated with a PS reflection on a CCP gather [1]. 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 ${\displaystyle v_{NMO}={\sqrt {\alpha \beta }}}$ [2]. 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.

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.

 ${\displaystyle x_{P}={\frac {\gamma }{1+\gamma }}x.}$ (73a)
 ${\displaystyle x_{P}={\frac {\sqrt {\gamma ^{2}+(\gamma ^{2}-1){\frac {x_{P}^{2}}{z^{2}}}}}{1+{\sqrt {\gamma ^{2}+(\gamma ^{2}-1){\frac {x_{P}^{2}}{z^{2}}}}}}}x.}$ (72c)
1. To begin with, note that the PP-wave velocity α can be estimated directly by velocity analysis of the PP data set itself.
2. We may assume an initial value for the velocity ratio γ = α/β and estimate an initial value for xP using equation (73a).
3. 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.
4. 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.
5. Use the γ(x, t0)-section and the PP-wave velocity α to calculate an updated value for xP(t0) from equation (72c).
6. 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

 ${\displaystyle t={\frac {1}{\alpha }}{\sqrt {x_{P}^{2}+{\frac {\alpha ^{2}\beta ^{2}}{(\alpha +\beta )^{2}}}t_{0}^{2}}}+{\frac {1}{\beta }}{\sqrt {(x-x_{p})^{2}+{\frac {\alpha ^{2}\beta ^{2}}{(\alpha +\beta )^{2}}}t_{0}^{2}}}.}$ (78)

By using the nonhyperbolic equation (78), follow the alternative procedure for PS-wave velocity analysis outlined below.

1. Again, estimate the PP-wave velocity α as before using the PP data set itself.
2. Also, assume an initial value for γ = α/β and estimate an initial value for xP using equation (73a).
3. 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.
4. Pick a function β(t0) at each CCP analysis location along the line over the survey area and derive a β(x, t0)-section.
5. Use the β(x, t0)-section and the PP-wave velocity α to calculate an updated value for xP(t0) from equation (72c).
6. 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 ${\displaystyle t_{0}^{(PS)}}$

 ${\displaystyle t_{0}^{(PS)}=\left({\frac {1}{\alpha }}+{\frac {1}{\beta }}\right)z.}$ (79a)

For the same depth z, the PP two-way time ${\displaystyle t_{0}^{(PP)}}$ is given by

 ${\displaystyle t_{0}^{(PP)}={\frac {2}{\alpha }}z.}$ (79b)

Now eliminate z between the two equations to get the relation between the PP and PS zero-offset times

 ${\displaystyle t_{0}^{(PP)}={\frac {2\beta }{\alpha +\beta }}t_{0}^{(PS)},}$ (80a)

or in terms of γ = α/β

 ${\displaystyle t_{0}^{(PP)}={\frac {2}{\gamma +1}}t_{0}^{(PS)}.}$ (80b)

So, α in equation (78) is specified at time ${\displaystyle t_{0}^{(PP)}}$ given by equation (80a), and a in equation (76) is specified at time ${\displaystyle t_{0}^{(PP)}}$ given by equation (80b).

## References

1. Tessmer and Behle, 1988, Tessmer, G. and Behle, A., 1988, Common reflection point data stacking technique for converted waves: Geophys. Prosp., 36, 671–688.
2. 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.
3. 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.