Radon-transform multiple attenuation
Consider the synthetic CMP gather in Figure 6.4-7a and the velocity-stack gather (Figure 6.4-7b) estimated from it using the discrete Radon transform. By including the entire velocity-stack gather in the summation in equation (10b), we get the fully reconstructed CMP gather shown in Figure 6.4-7c. That result is reproduced in Figure 6.4-19a. Aside from the loss of high-frequency energy at early times, this modeled CMP gather is a close approximation to the original CMP gather (Figure 6.4-7a).
Instead of including the entire velocity-stack gather (Figure 6.3-15b) in the summation in equation (10b), a CMP gather with only multiples (Figure 6.4-19b) or only primaries (Figure 6.4-19c) can be reconstructed by simply assigning suitable pass-reject zones over the velocity-stack gather. Compare the modeled multiples-only and primaries-only CMP gathers (Figures 6.4-19b,c) with the actual CMP gathers shown in Figures 6.4-2b,a. (The modeled shallow primary in Figure 6.4-19c corresponds to the primary in Figure 6.4-2b.) It appears that, although insignificant, the multiples-only gather (Figure 6.4-19b) contains some residual primary energy, and the primaries-only gather (Figure 6.4-19c) contains some residual multiple energy. In practice, it often is desirable to model the multiples and subtract the result from the actual CMP gather . One reason for this is the necessity to retain in CMP data some of the nonhyperbolic energy, such as diffractions. In the present example, Figure 6.4-19d shows the difference between the original CMP gather (Figure 6.4-7a) and the modeled multiples-only CMP gather (Figure 6.4-19b). When compared with Figure 6.4-19c, the subtraction result shown in Figure 6.4-19d shows differences at early times due to the t2-stretching artifacts.
Figure 6.4-2 (a) A synthetic CMP gather with three primary reflections; (b) a synthetic CMP gather with one primary reflection (arrival time at 0.2 s at zero-offset time) and its multiples; (c) composite CMP gather containing the primaries and multiples in (a) and (b); (d) the conventional velocity-stack gather derived from the composite CMP gather using equation (10a). Note the amplitude smearing along the velocity axis.
Figure 6.4-7 (a) Synthetic CMP gather; (b) velocity-stack gather based on the discrete Radon transform; (c) reconstructed CMP gather.
Figure 6.4-8 (a) The synthetic CMP gather as in Figure 6.4-2c; (b) the same gather with added band-limited random noise; (c) the conventional velocity-stack gather (b); (d) the discrete Radon transform of (b). Note the improved velocity resolution in (d) as compared to the amplitude smearing in (c).
Figure 6.4-20 Reconstruction of the noise-contaminated CMP gather in Figure 6.4-8b using (a) the entire velocity-stack gather in Figure 6.4-8d; (b) allowing only the multiple energy; (c) allowing only the primary energy; (d) subtraction of (b) from Figure 6.4-8b. Aside from the muted zone, the noise in the original gather (Figure 6.4-8b) is retained.
The subtraction actually tends to retain the original texture of the data; this is demonstrated with the noise-contaminated CMP gather in Figure 6.4-8b. Using the entire velocity-stack gather (Figure 6.4-8d) associated with this CMP gather, we get the fully reconstructed CMP gather shown in Figure 6.4-20a. Note that this modeled CMP gather is a close approximation to the noise-free CMP gather (Figure 6.4-7a). Also, compare the modeled multiples-only and primaries-only CMP gathers (Figures 6.4-20b,c) with the corresponding results from the noise-free CMP gather (Figures 6.4-19b,c). Figure 6.4-20d shows the difference between the original CMP gather (Figure 6.4-8b) and the modeled multiples-only CMP gather (Figure 6.4-20b). When compared with Figure 6.4-20c, the subtraction result shown in Figure 6.4-20d retains the original noise component present in the data (Figure 6.4-8b).
Now consider a field data example for separation of primaries and multiples by the discrete Radon transform. Shown in Figure 6.4-21 are the deep-water CMP gather, and the reconstructed primaries-only, multiples-only, and the subtraction gathers. Corresponding velocity spectra in Figure 6.4-22 clearly show that multiples largely have been removed from the input CMP gather. Nevertheless, a complete separation of multiples from primaries is not achievable. As an example, note the residual primary energy especially visible above 4 s in the multiples-only CMP gather (Figure 6.4-21c). Accordingly, the subtraction result (Figure 6.4-21d) inevitably will have some remnant multiple energy. This is especially apparent in the corresponding velocity spectrum (Figure 6.4-22d); note the small coherency peaks in the multiple zone below 4 s.
Since the Radon-transform multiple attenuation using velocity-stack gathers exploits the velocity discrimination between primaries and multiples, it is appropriate to compare the technique with other methods that also are based on the same criterion. Figure 6.4-23 shows the synthetic CMP gather with multiple attenuation using the Radon transform, model-based (multiple attenuation in the CMP domain) and frequency-wavenumber filtering (frequency-wavenumber filtering) methods. Note that the model-based method fails to preserve the amplitude characteristics of the input data (Figure 6.4-23c). This is a direct consequence of the problems in creating model traces for multiples as was referred to in multiple attenuation in the CMP domain. The frequency-wavenumber filtering method has caused attenuation of priamries at near offsets — an effect similar to inside-trace muting (Figure 6.4-23d). Among the three approaches, the Radon transform method appears to best preserve amplitude and phase characteristics of the input data (Figure 6.4-23b).
Further comparisons between the three methods based on velocity discrimination can be made using the noise-contaminated CMP gather in Figure 6.4-24. Again, the Radon transform yields the most desirable result. In fact, whenever data, which require multiple attenuation, are used for amplitude inversion to estimate acoustic impedance or amplitude variation with offset (AVO) analysis, the preferred technique for multiple attenuation most often is the Radon transform.
A challenging data example with short-period interbed multiples is shown in Figure 6.4-25. The velocity spectra computed from the original CMP gather (Figure 6.4-25a) and the gather with multiples removed (Figure 6.4-25d) are shown in Figure 6.4-26. Portions of CMP stacked sections with and without velocity-stack processing for multiple suppression are shown in Figure 6.4-27. An important observation in Figure 6.4-27a is the apparent lateral continuity caused by the multiples. This continuity is replaced, in Figure 6.4-27b, with features that are perhaps geologically more detailed and plausable. Note the presence of a subtle structural closure at 1.5 s in Figure 6.4-27b; this feature is completely disguised among the multiples in Figure 6.4-27a. Unfortunately, because of the unavailability of well logs, no definite assessment can be made about the details in the CMP stacked section processed for multiple suppression (Figure 6.4-27b).
We now demonstrate application of the Radon transform to moveout-corrected CMP gathers . Figures 6.4-28 shows a CMP stack without multiple attenuation. Water-bottom and peg-leg multiples dominate the lower half of the sections and interfere with primary reflections of interest. Strong multiple reflections also are seen on the selected CMP gather in Figure 6.4-29a. Most of the multiples — water-bottom and the peg-legs associated with the depositional sequence boundary at 2 s, are long-period.
Hampson’s implementation of the Radon transform requires input CMP gathers to be moveout-corrected using a primary velocity function (Figure 6.4-29b). The aim is to make the moveout of events — primaries and multiples, approximately parabolic. Prior to Radon transformation, spatial interpolation of the data may be needed to make the trace interval sufficiently small (Figure 6.4-29c). The Radon transformation itself is done using the moveout at a reference offset, instead of velocity, as the variable for the horizontal axis (Figure 6.4-29d).
Multiple attenuation in the transform domain is achieved by rejecting a zone that includes the primaries (Figure 6.4-30a). The inverse transform yields the reconstructed gather that contains presumably only multiples (Figure 6.4-30b). Again, to preserve data characteristics, rather than modeling the primaries by reconstruction, it is preferred to model the multiples (Figure 6.4-30b) and subtract the modeled gather from the original (Figure 6.4-29c). The difference gather should contain the primaries (Figure 6.4-30c). Finally, traces which were generated during trace interpolation (Figure 6.4-29c) are dropped (Figure 6.4-30d). Compare Figure 6.4-30d — the gather after multiple attenuation, with Figure 6.4-29b — the gather before multiple attenuation. Aside from a residual of the water-bottom multiples, much of the energy associated with multiples has been removed.
Figure 6.4-24 (a) Synthetic CMP gather as in Figure 6.4-2c with added band-limited random noise; after multiple attenuation using (b) the Radon transform, (c) the modeling of multiples in t − x domain (multiple attenuation in the CMP domain), and (d) the f − k method (frequency-wavenumber filtering).
Figure 6.4-27 (a) A portion of the CMP stacked section associated with the CMP gather in Figure 6.4-25a with short-period multiples; (b) same portion of the CMP stacked section associated with the CMP gather in Figure 6.4-25d with velocity-stack processing for multiple suppression. Note the apparent lateral continuity caused by the short-period multiples in (a); this false continuity is removed in (b), thereby uncovering a probable subtle structural feature at 1.5 s below midpoint A. (Data courtesy Abu Dhabi National Oil Company.)
Figure 6.4-28 A CMP-stacked section without multiple attenuation. (Data courtesy BP-Amoco and Shell.)
Figure 6.4-30 (a) The Radon transform as in Figure 6.4-29d after muting the primary zone; (b) the reconstructed gather that contains only multiples; (c) the difference gather obtained by subtracting (b) from Figure 6.4-29c; (d) the same gather as in (c) but retaining every other trace with offsets as in Figure 6.4-29b. Compare (d) — the gather after multiple attenuation, with Figure 6.4-29b — the gather before multiple attenuation.
Figure 6.4-31 The CMP-stacked section as in Figure 6.4-28 after multiple attenuation.
Figure 6.4-32 The CMP-stacked section as in Figure 6.4-31 after migration.
Figure 6.4-31 shows the CMP stack after Radon-transform multiple attenuation. Note that multiple attenuation has uncovered the primary the event at 3.5 s associated with an unconformity. Multiple attenuation also helps migration to yield an improved image as shown in Figure 6.4-32.
- Hampson (1986), Hampson, D., 1986, Inverse velocity stacking for multiple elimination: J. Can. Soc. Expl. Geophys., 22, 44–55.
- Velocity-stack transformation
- The discrete Radon transform
- The parabolic Radon transform
- Practical considerations
- Impulse response of the velocity-stack operator
- Field data examples