# Common-conversion-point binning

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

We learned in analysis of amplitude variation with offset that an incident P-wave is partitioned at a layer boundary into reflected and transmitted P- and S-wave components. Consider the raypath geometry in Figure 11.6-30a for an incident P-wave generated by the source S1 and a flat reflector. The reflection angle for the PP-wave is equal to the angle of incidence; however, the reflection angle for the PS-wave is smaller than the angle of incidence. As a result, the PP reflection will follow a symmetric raypath and be recorded at receiver location R2, while the PS reflection will follow an asymmetric raypath and be recorded at receiver location R1.

Now consider the common-midpoint (CMP) raypath geometry for a source-receiver pair S1R1 shown in Figure 11.6-30b. There are two reflection arrivals at the receiver location R1 associated with the PP and PS raypaths. The reflection point B at which the incident P-wave is converted to the S-wave is displaced in the lateral direction by some distance d away from the reflection point A at which the incident P-wave is reflected and recorded at the same receiver location R1 as the converted S-wave. This means that, for an earth model with flat layers, the PP-wave reflection points coincide with the midpoint location (Figure 11.6-31a); whereas, the PS conversion points do not (Figure 11.6-31b). As a direct consequence of this observation, the notion of a CMP gather based on sorting PP data from acquisition coordinates — source and receiver, to processing coordinates — midpoint and offset, such that traces in the gather have the same midpoint coordinate, is not applicable to PS data. Instead PS data need to be sorted into common-conversion-point (CCP) gathers such that traces in this gather have the same conversion point coordinate.

An important aspect of CCP sorting is that the asymmetric raypath associated with the PS reflection gives rise to a periodic variation in fold of the CCP gathers. As for the conventional P-wave data with variations in fold caused by irregular recording geometry, amplitudes of the stacked PS data are adversely affected by the variation in the CCP fold [1] [2]. Just as one resorts to flexible bin size in the processing of 3-D seismic data to accommodate variations in fold, the same strategy may be applied for the PS data processing.

Binning the PS data into CCP gathers requires knowledge of the conversion-point coordinate xP. Referring to Figure 11.6-31b, note that the conversion-point coordinate follows a trajectory indicated by the broken curve that, in general, depends on the reflector depth [3].

To derive an expression for xP, refer to the geometry of the PS-raypath shown in Figure 11.6-32. By Snell’s law, we know that

 ${\displaystyle {\frac {\sin \varphi _{0}}{\alpha }}={\frac {\sin \psi _{1}}{\beta }},}$ (70)

where α and β are the P-wave and S-wave velocities, respectively, and φ0 is the P-wave angle of incidence and ψ1 is the reflection angle for the converted S-wave.

From the geometry of Figure 11.6-32, note that

 ${\displaystyle \sin \varphi _{0}={\frac {x_{P}}{\sqrt {x_{P}^{2}+z^{2}}}}}$ (71a)

and

 ${\displaystyle \sin \psi _{1}={\frac {x_{S}}{\sqrt {x_{S}^{2}+z^{2}}}},}$ (71b)

where xP and xS are the lateral distances from the CCP to the source and receiver locations, respectively. Substitute equations (71a,71b) into equation (70), square and rearrange the terms to get

 ${\displaystyle {\frac {x_{S}^{2}}{x_{P}^{2}}}={\frac {\beta ^{2}}{\alpha ^{2}}}{\frac {x_{S}^{2}+z^{2}}{x_{P}^{2}+z^{2}}}.}$ (72a)

Apply some algebraic manipulation to solve equation (72a) for xS

 ${\displaystyle x_{S}={\frac {x_{P}}{\sqrt {\gamma ^{2}+(\gamma ^{2}-1){\frac {x_{P}^{2}}{z^{2}}}}}},}$ (72b)

where γ = α/β. Finally, substitute the relation xS = x − xP, where x is the source-receiver offset, into equation (72b) to get the desired expression

 ${\displaystyle x_{P}={\frac {\sqrt {\gamma ^{2}+(\gamma ^{2}-1){\frac {x_{P}^{2}}{z^{2}}}}}{1+{\sqrt {\gamma ^{2}+(\gamma ^{2}-1){\frac {x_{P}^{2}}{z^{2}}}}}}}x.}$ (72c)

From Figure 11.6-31b, note that the CCP location moves closer to CMP location as the depth of the reflector increases. At infinite depth, the CCP location reaches an asymptotic conversion point (ACP) [4]. In the limit z → ∞, equation (72c) gives the ACP coordinate xP with respect to the source location

 ${\displaystyle x_{P}={\frac {\gamma }{1+\gamma }}x.}$ (73a)

Since β < α, the conversion point is closer to the receiver location than the source location (Figure 11.6-31b). The displacement d = xP − x/2 of the asymptotic conversion point from the midpoint is, by way of equation (73a),

 ${\displaystyle d={\frac {1}{2}}\left({\frac {\gamma -1}{\gamma +1}}\right)x.}$ (73b)

While CCP binning may be performed using the ACP coordinate given by equation (73a), more accurate binning techniques account for the depth-dependence of the CCP coordinate xP based on a solution to equation (72c) [3] [5]. Because of the quartic form of equation (72c) in terms of xP, an iterative solution may be preferred in practice [5] [6] [7]. The iteration may be started by substituting the asymptotic form of xP given by equation (73a) into the right-hand side of equation (72c). The new value of xP may then be back substituted into equation (72c) to continue with the iteration.

Whatever the estimation procedure, note from equation (72c) that xP depends both on depth to the reflector and the velocity ratio γ = α/β. Unless a value for the velocity ratio is assumed, it follows that CCP binning requires velocity analysis of PS data to determine the velocity ratio γ. Additionally, an accurate CCP binning strictly requires the knowledge of reflector depths; thus, the advocation of an implicit requirement that 4-C seismic data analysis should be done in the depth domain. This requirement may be waivered if we only consider a horizontally layered earth model as in the next subsection.

## References

1. Eaton and Lawton, 1992, Eaton, D. W. S. and Lawton, D. C., 1992, P-SV stacking charts and binning periodicity: Geophysics, 57, 745–748.
2. Li and Yuan, 1999, Caldwell, J., 1999, Marine multicomponent seismology: The Leading Edge, 1274–1282.
3. Tessmer and Behle, 1988, Tessmer, G. and Behle, A., 1988, Common reflection point data stacking technique for converted waves: Geophys. Prosp., 36, 671–688.
4. 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.
5. Zhang and Robinson, 1992, Zhang, Y. and Robinson, E. A., 1992, Stacking P-SV converted wave data with raypath velocity: 62nd Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1214–1217.
6. Zhang, 1996, Zhang, Y., 1996, Nonhyperbolic converted wave velocity analysis and normal moveout: 66th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1555–1558.
7. Yuan and Li, 1997, Yuan, J. and Li., X-Y., 1997, Converted-wave CCP binning and velocity analysis: in Processing three-component seafloor seismic data, Edinborough University.