Slant-stack multiple attenuation
The slant-stack multiple attenuation technique is based on prediction of multiples. Alam and Austin  and Treitel  investigated the application of predictive deconvolution in the slant-stack domain for multiple attenuation. The application of predictive deconvolution to multiple attenuation is valid strictly for vertical incidence and the zero-offset case. Multiples are not periodic at nonzero offsets. Figure 6.3-21 shows a sketch of a shot gather with primary P (water-bottom reflection) and its multiples M1, M2 with the corresponding slant-stack gather. The time separations between the multiple arrivals at a particular offset x0 are equal only if x = 0.
Taner  first recognized that the time separations between the arrivals are equal along a radial direction OR. A trace can be constructed by extracting the samples along one of these radial directions. The angle of propagation is constant along this radial trace . A radial trace in a layered medium is called a Snell trace . In a layered medium, the Snell trace would not follow a straight path as in Figure 6.3-21, since its angle of propagation changes at layer boundaries according to Snell’s law (Figure 6.3-6).
Taner  applied predictive deconvolution along radial traces to successfully eliminate long-period multiples. Note that the magnitude of the time separations between multiples is different from one radial trace to another (Figure 6.3-21). However, the time separations are equal along each of the slanted paths of summation. Therefore, a predictive deconvolution operator can be designed from the autocorrelogram of each p trace (such as that denoted by p0 in Figure 6.3-21) and applied to suppress multiples. This is demonstrated in Figure 6.3-22. The synthetic shot gather contains a water-bottom reflection and its multiples (Figure 6.3-22a). Note that the periodic nature of the multiples is not apparent on the autocorrelogram. Therefore, predictive deconvolution should not be expected to do well in attenuating these multiples when applied to the shot gather.
The shot gather in Figure 6.3-22a now is transformed to the slant-stack domain. Figures 6.3-22c and 6.3-22e show the slant-stack gather before and after predictive deconvolution was applied. Figure 6.3-22g shows reconstruction of the shot gather from the slant-stack gather in Figure 6.3-22e. Autocorrelograms before and after deconvolution in the slant-stack domain are shown beneath the respective panels. Unlike in the autocorrelogram of the shot gather in Figure 6.3-22b, the periodic nature of the multiples in the data is pronounced in the autocorrelogram of the slant-stack gather (Figure 6.3-22d). Note that the periodicity of multiples changes from one p trace to the next. The largest period occurs along the trace that corresponds to the minimum p value. The autocorrelogram after predictive deconvolution shows that the energy in the lags less than the specified prediction lag is retained, while the multiple energy is attenuated (Figure 6.3-22f).
Figure 6.3-21 The periodicity of multiples along radial trace OR and down the p traces.
Figure F-1 (a) A shot gather containing the strong reflected and refracted multiples associated with hard water-bottom conditions. Here, CC′ = critical-angle energy. (b) The slant-stack gather derived from this shot gather. (c) The ω − p gather derived from the τ − p gather in panel (b). The inverse of p is the horizontal phase velocity. This figure demonstrates the dispersive nature of guided waves; that is, phase velocity is a function of frequency for all propagating normal mode components. These modes are represented by the curved trajectories on panel (c).
Prediction lag α and operator length n must be specified by examining the autocorrelogram of the slant-stack gather (Figure 6.3-22d). These two parameters are specified for the trace corresponding to the lowest p value, as indicated in Figure 6.3-22d. Operator length is kept constant, while prediction lag is adjusted based on the p value across the gather :
where α(0) = prediction lag at p = 0 and vw is the velocity of the primary reflection, the multiples of which are targeted for attenuation. At higher p values, the prediction lag decreases. Compare the reconstructed offset gather (Figure 6.3-22g) with the input gather (Figure 6.3-22a), and note that the output contains the water-bottom primary (the only one present in the input data) and a residual of the first multiple.
Multiple attenuation in the slant-stack domain is demonstrated further by the synthetic data in Figure 6.3-23a. The synthetic shot record in Figure 6.3-23a is a simulation of the shot gather in Figure F-1a using normal-mode modeling (Section F.1). Several arrivals are identified: C is the direct arrival; A is the refracted arrival associated with the hard water bottom; B is the water-bottom reflection; M1, M2, and M3 are the refracted multiples; and m1, m2 and m3 are the reflected multiples. D is an artifact of the normal-mode modeling technique (Section F.1).
Figure 6.3-23c is the slant-stack gather of the synthetic shot record in Figure 6.3-23a. This gather should be compared with the slant-stack gather of the field data in Figure F-1b. Refraction A and its multiples M1, M2, and M3 map onto points in the slant-stack domain. Figure 6.3-23d is the autocorrelogram of the τ − p gather. Unlike the autocorrelogram of the offset data (Figure 6.3-23b), note that it exhibits the periodic nature of the multiples in the data. After applying predictive deconvolution, the slant-stack gather in Figure 6.3-23e results. Only the refracted arrival A and the water-bottom reflection B remain. The nearly linear streaks, which also are present in the unprocessed slant-stack gather (Figure 6.3-23c), are artifacts caused by the finite cable length. The autocorrelogram after deconvolution is free from the multiple energy (Figure 6.3-23f). The prediction lag and operator length for the minimum p-value are as labeled in Figure 6.3-23d. Adjustment for the prediction lag was made across the gather using equation (8). Finally, reconstruction of the shot gather is shown in Figure 6.3-23g. When compared with Figure 6.3-23a, note that both the refracted and reflected primaries are retained, while their associated multiples are largely attenuated.
The performance of slant-stack multiple attenuation on field data now is examined. Figure 6.3-24a shows a shot gather that contains a strong water-bottom reflection A, two distinct primaries B and C, the water-bottom multiples D and E, and the peg-leg F, which is associated with the primary event B. The slant-stack gathers before and after predictive deconvolution are shown in Figures 6.3-24b and 6.3-24d with their respective autocorrelograms (Figures 6.3-24c and 6.3-24e). Note that multiples are significantly attenuated in the reconstructed gather (Figure 6.3-24f).
Choice of the prediction lag and operator length is tricky for this particular data set. From the autocorrelogram in Figure 6.3-24c, note the energy G, which is caused by the correlation of two primaries — A and B in Figure 6.3-24a. Energy H is caused by the correlation of the water-bottom multiples. Prediction lag is chosen to retain primary energy G, and the operator length is chosen to include the multiple energy H. Note that in Figure 6.3-24e multiple energy H is significantly attenuated and primary energy G is preserved.
Figure 6.3-22 Multiple attenuation in the slant-stack domain. (a) A shot gather; (b) its autocorrelogram; (c) the slant-stack gather; (d) the autocorrelogram of (c); (e) the slant-stack gather after predictive deconvolution, where operator length = 240 ms and prediction lag at p = 120 ms; (f) the autocorrelogram of (e); (g) reconstruction of the shot gather from (e).
Figure 6.3-23 (a) Simulation of the shot gather shown in Figure F-1a by normal-mode modeling; (b) autocorrelogram of this synthetic gather; (c) slant stack of the synthetic gather; (d) autocorrelogram of the slant-stack gather; (e) the slant-stack gather in (c) after predictive deconvolution; (f) the autocorrelogram of (e); (g) reconstruction of the synthetic gather from the slant-stack gather in (e). (Refer to the text for a description of the labeled events.
Figure 6.3-24 Multiple attenuation in the slant-stack domain. (a) A field record without geometric spreading correction; (b) the slant-stack gather obtained from it; (c) the autocorrelogram of (b); (d) the slant-stack gather after predictive deconvolution, where operator length = 400 ms and prediction lag (at p = 0) = 700 ms; (e) the autocorrelogram of (d); (f) reconstruction of the shot gather from (d).
Since slant stacking is a plane-wave decomposition, and since plane waves do not have spherical divergence, input to slant stacking must not be compensated for by geometric spreading. Preserving correct amplitude relationships is essential for the effectiveness of slant-stack multiple attenuation. The geometric spreading correction is applied to offset data by using a primary velocity function. This enhances the multiples in the data and destroys the amplitude relationship between them. Predictive deconvolution in the offset domain then may not suppress these multiples effectively.
After multiple attenuation in the slant-stack domain, reconstruction of the offset data is performed, the geometric spreading correction is applied, and processing is continued. Figures 6.3-25a and 6.3-25b show the shot gathers in Figures 6.3-24f and 6.3-24a after the geometric spreading correction. Note that strong multiples are attenuated significantly after deconvolution in the slant-stack domain. While the autocorrelogram of the shot gather with slant-stack processing (Figure 6.3-24c) exhibits the presence of strong multiples, the autocorrelogram of the shot gather without slant-stack processing does not exhibit the periodicity of multiples (Figure 6.3-25c).
Figures 6.1-2, 6.1-3, and 6.1-5 show that when multiples are in the form of short-period reverberations, the autocorrelogram of the offset data seems to adequately represent the periodic nature of multiples. Hence, predictive deconvolution of the offset data often can remove reverberations. On the other hand, long-period multiples are poorly represented by the autocorrelogram of the offset data (Figure 6.3-25c) and are better defined in the slant-stack domain (Figure 6.3-24c).
Figure 6.3-26 shows the stacked section that corresponds to the field data in Figure 6.3-25b. The slant-stack processed section, which corresponds to the data in Figure 6.3-25a, is shown in Figure 6.3-27. Major primary reflections stand out more distinctively in the slant-stack processed section.
Figure 6.1-2 Same gathers as in Figure 6.1-1 after t-squared scaling for geometric spreading correction and muting guided waves.
Figure 6.1-3 Same gathers as in Figure 6.1-2 after predictive deconvolution.
Figure 6.1-5 A field record containing short-period reverberations before (a) and after deconvolution (b). The solid lines represent the start and end times for the autocorrelation estimation windows.
- Alam and Austin (1981), Alam, A. and Austin, J., 1981, Multiple attenuation using slant stacks: Tech. Rep., Western Geophysical Company.
- Treitel et al. (1982), Treitel, S., Gutowski, P. R., and Wagner, D. E., 1982, Plane-wave decomposition of seismograms: Geophysics, 47, 1375–1401.
- Taner (1980), Taner, M. T., 1980, Long-period sea-floor multiples and their attenuation: Geophys. Prosp., 28, 30–48.
- Ottolini, 1982, Ottolini, R., 1982, Migration of seismic data in angle-midpoint coordinates: Ph.D. thesis, Stanford University.
- Claerbout, 1985, Claerbout, J. F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Physical aspects of slant stacking
- Slant-stack transformation
- Practical aspects of slant stacking
- Slant-stack parameters
- Time-variant dip filtering