# Frequency-wavenumber multiple attenuation

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 |

Figure 6.2-11a shows the synthetic CMP gather from Figure 6.1-9c and its 2-D amplitude spectrum. The primary and multiple energy can be separated into two different quadrants in the *f − k* plane. This is achieved by NMO correcting the gather using a velocity function (labeled as *V B* in Figure 6.1-9d) that is between the primary and multiple velocities. The resulting NMO-corrected gather and its 2-D amplitude spectrum are shown in Figure 6.2-11b. The multiples are undercorrected, while the primaries are overcorrected. In the *f − k* domain, the multiples and primaries for the most part map onto two different quadrants (labeled as *P* for primaries and *M* for multiples). The exception to this separation is the near-offset energy (both primary and multiple), which almost entirely maps along the frequency axis. This occurs because multiples and primaries have no significant moveout difference at near offsets. Aliased energy (such as *A*) is wrapped around and mapped to the wrong quadrant. (Spatial aliasing is discussed in detail in the 2-D Fourier transform.)

Multiples can be attenuated by zeroing the quadrant corresponding to multiple energy in the *f − k* domain (Figure 6.2-11c) ^{[1]}^{[2]}. However, note that spatially aliased multiple energy remains in the gather (labeled as *A* in Figure 6.2-11c). A shown in Figure 6.2-11d, besides zeroing out the multiple quadrant, a reject zone labeled as *R* can be imposed on the primary quadrant. The *f − k* filtered CMP gather (Figure 6.2-11d) now also is free of the aliased energy (compare with Figure 6.2-11c). Inverse NMO correction (Figure 6.2-11e) using the same intermediate velocity function *V B* (Figure 6.1-9d) restores the original moveout of the primaries. Following this procedure, apply NMO correction using the primary velocity function *V P* (Figure 6.1-9d) as shown in Figure 6.2-11f. The stacked trace of this gather displayed repeatedly exhibits very little multiple energy (Figure 6.2-11g).

**Figure 6.2-11**(a) The synthetic CMP gather same as in Figure 6.1-9c; (b) after NMO correction using a velocity function (labeled as*V B*in Figure 6.1-9d) between the multiple and primary trend; (c) the result of zeroing the*f − k*quadrant associated with multiples; (d) the same as (c), except that in addition to zeroing the left quadrant, a portion of the right quadrant in the*f − k*spectrum (denoted by*R*) also is zeroed to suppress aliased energy; (e) the result of applying inverse NMO correction to (a) using the velocity function labeled as*V B*in Figure 6.1-9d; (f) the result of applying NMO correction to (e) using the primary velocity function labeled as*V P*in Figure 6.1-9d; (g) stack of (c) repeated to emphasize the strong events. The bottom panels show the corresponding*f − k*spectra.**Figure 6.2-12**(a) The CMP gathers in Figure 6.1-8a after NMO correction using a velocity function (labeled as*V B*in Figure 6.1-8b) between the primary and multiple velocities; (b) velocity spectrum at CMP 186 estimated from the*f − k*dip-filtered gather shown to the left of the spectrum. Compare this with Figure 6.1-8b. (c) The same CMP gathers as in (a) after*f − k*multiple attenuation followed by NMO correction using the primary velocities derived from velocity spectrum (b). (d) The CMP stack derived from the CMP gathers as in (c) after*f − k*multiple attenuation.**Figure 6.1-8**(a) Three CMP gathers with strong multiples; (b) velocity analysis at CMP 186, where*V P*= primary velocity trend,*V M*1 = slow (water-bottom) multiples, and*V M*2 = fast (peg-leg) multiples. (*V B*is the velocity function used in generating Figure 6.2-15a.) For reference, the CMP gather is displayed next to the velocity spectrum. (c) The same CMP gathers as in (a) after NMO correction using the primary velocities. (d) CMP stack using the gathers as in (c). (Data courtesy Petro-Canada Resources.)**Figure 6.1-9**Synthetic CMP gathers containing (a) primaries, (b) water-bottom multiples, (c) superposition of (a) and (b). (d) The velocity spectrum derived from (c). Here,*W*= water-bottom primary,*V M*= velocity function for multiples,*V P*= velocity function for primaries,*V B*= a velocity function between*V M*and*V P*used in generating Figure 6.2-12b.

The sequence for *f − k* filtering of multiple attenuation is as follows:

- Apply moveout correction to CMP gathers using a velocity function
*v*such that_{b}*v*<_{m}*v*<_{b}*v*, where_{p}*v*and_{m}*v*are velocity functions associated with multiples and primaries, respectively._{p} - Apply 2-F Fourier transform.
- Zero the quadrant associated with the multiples and, if required, the zone that contains the energy associated with the aliased multiples within the primaries quadrant.
- Apply inverse 2-D Fourier transform.
- Apply inverse moveout correction using the velocity function
*v*as in step (a)._{b} - Perform velocity analysis to update the picks for primary velocity functions.

Now consider the field data example shown in Figure 6.2-12. The moveout difference between the primaries and multiples is apparent in the CMP gathers (Figure 6.1-8a). To the left of the primary velocity function labeled as *V P* in Figure 6.1-8b, all peaks are associated with water-bottom and peg-leg multiples. Apply moveout correction using a velocity function *V B* that lies between the primary and multiple velocities. As a result, the multiples are undercorrected and the primaries are overcorrected as shown in Figure 6.2-12a. Now consider the moveout of primaries and multiples in the *f − k* domain. The primary and multiple indicated in Figure 6.1-8a map to the same quadrant (say positive quadrant) in the *f − k* domain before moveout correction. The same events after moveout correction using an intermediate velocity map into two different quadrants; in particular, the multiple maps into the positive quadrant and the primary maps into the negative quadrant. Thus, by zeroing one quadrant in which the multiples are clustered, the primaries can be enhanced.

A CMP gather and the associated velocity spectrum following *f − k* multiple attenuation are shown in Figure 6.2-12b. When compared with Figure 6.1-8b, Figure 6.2-12b shows that the energy in the multiples region in the velocity spectrum was attenuated, while the primary velocity trend was enchanced. Finally, Figure 6.2-12c shows selected CMP gathers after moveout correction using the primary velocities picked from the velocity spectrum in Figure 6.2-12b. Stacking of these gathers yields the section in Figure 6.2-12d. This section should be compared with Figures 6.1-8d.

**Figure 6.2-13**A CMP stack with no multiple attenuation, including pre- and poststack deconvolution.**Figure 6.2-14**The CMP stack as in Figure 6.2-13 with prestack deconvolution,*f − k*filtering for multiple attenuation, and poststack deconvolution.**Figure 6.2-16**The CMP stack as in Figure 6.2-13 with prestack deconvolution.**Figure 6.2-17**The CMP stack as in Figure 6.2-13 with pre- and poststack deconvolution.**Figure 6.2-18**The CMP stack as in Figure 6.2-13 with prestack deconvolution and*f − k*filtering for multiple attenuation.

In practice, there are variations in selecting the velocity function used to apply moveout correction prior to *f − k* filtering. An alternative strategy is to apply NMO correction using the multiple velocity, then zero the energy along the frequency axis in addition to that in the multiple quadrant of the *f − k* spectrum. Another strategy is to apply NMO correction using primary velocities and place a tight pass-zone around the frequency axis. Finally, note that the *f − k* method of velocity discrimination is one type of *f − k* filtering. Thus, we must deal with the same practical issues discussed in the 2-D Fourier transform — in particular, wraparound, spatial aliasing, and tapering over the boundary between the pass and reject zones.

Multiples are best attenuated when a combination of two methods based on periodicity of multiples and velocity discrimination between primaries and multiples are used. Figure 6.2-13 shows a CMP-stacked section with no attempt made to attenuate the multiples. Specifically, the processing sequence did not include deconvolution nor *f − k* filtering. Note the abundance of peg-leg multiples associated with the strong reflections; these multiples dominate the section below 2 s.

By combining prestack deconvolution with *f − k* filtering followed by poststack deconvolution, as demonstrated in Figure 6.2-14, multiples in Figure 6.2-13 are largely attenuated despite the complicated nature of the reflectors that originate the peg-legs. The class of multiples present in the stacked section of Figure 6.2-13 is clearly identified in the corresponding velocity spectrum shown in Figure 6.2-15. Note that much of the energy associated with the multiples is absent in the velocity spectrum shown in Figure 6.2-15 corresponding to the stacked section in Figure 6.2-14. Compare the velocity spectra in Figure 6.2-15 computed from the CMP gather without and with multiple attenuation and note that, following multiple attenuation, the energy of the primaries has been enhanced relative to that of the multiples.

Prestack deconvolution alone has limited effect in attenuating multiples (Figure 6.2-16). When combined with poststack deconvolution, however, it often is successful in attenuating a large class of multiples (Figure 6.2-17). The key to effective multiple attenuation by predictive deconvolution is to be generous with the operator length. As for the example shown in Figure 6.2-17, an operator length as long as 480 ms may need to be considered.

A combination of prestack deconvolution and *f − k* filtering for multiple attenuation yields the section shown in Figure 6.2-18. Compare with Figure 6.2-17 and note the more effective multiple attenuation by combining deconvolution with *f − k* filtering. The addition of poststack deconvolution to the sequence that includes prestack deconvolution and *f − k* filtering yields the most favorable result within the context of multiple attenuation (Figure 6.2-14).

## References

## See also

- Random noise and frequency-wavenumber filtering
- Statics corrections and frequency-wavenumber filtering
- Dip filtering of coherent linear noise