# Velocity analysis of PS data

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 |

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* = *x _{P}* = 0 in equation (

**74**) to see that the two-way zero-offset

*PS*traveltime is given by

**(**)

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

**(**)

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 ^{[1]}. In equation (**77**), *t* and *t*_{0} mean the same traveltimes as in equation (**76**), and *v _{NMO}* 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

^{[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, *v _{NMO}*, 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

*x*. But, in practice, we do not have to scan for all three parameters. Instead, the iterative procedure described below may be followed.

_{P}

**(**)

**(**)

- 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*x*using equation (_{P}**73a**). - Knowing
*α*and*x*, use the nonhyperbolic moveout equation (_{P}**76**) to scan for*γ*as a function of*t*_{0}. Figure 11.6-34 shows a*γ*-spectrum computed from the CCP gather in Figure 11.6-33a. - Pick a function
*γ*(*t*_{0}) at each CCP analysis location along the line over the survey area and derive a*γ*(*x, t*_{0})-section as shown in Figure 11.6-35. - Use the
*γ*(*x, t*_{0})-section and the*PP*-wave velocity*α*to calculate an updated value for*x*(_{P}*t*_{0}) from equation (**72c**). - Substitute the updated
*x*(_{P}*t*_{0}) 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

**(**)

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

- 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*x*using equation (_{P}**73a**). - Knowing
*α*and*x*, use the nonhyperbolic moveout equation (_{P}**78**) to scan for*β*as a function of*t*_{0}. 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*(*t*_{0}) at each CCP analysis location along the line over the survey area and derive a*β*(*x, t*_{0})-section. - Use the
*β*(*x, t*_{0})-section and the*PP*-wave velocity*α*to calculate an updated value for*x*(_{P}*t*_{0}) from equation (**72c**). - Substitute the updated
*x*(_{P}*t*_{0}) 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.

**Figure 11.6-32**Geometry of a common-conversion-point (CCP) raypath used to derive the reflection traveltime equation (**74**) for the*PS*-wave.**Figure 11.6-20**A CMP gather from the*PP*-data as in Figure 11.6-14 and the semblance spectrum for*PP*-velocity analysis. (Processing by Orhan Yilmaz, 1999).

One subtle issue is related to the time at which a is specified in equations (**76**) and (**78**). The two-way zero-offset time *t*_{0} 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*

**(**)

So, *α* in equation (**78**) is specified at time given by equation (**80a**), and a in equation (**76**) is specified at time given by equation (**80b**).

## References

- ↑ 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. - ↑
^{3.0}^{3.1}^{3.2}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.

## See also

- 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