# Karhunen-Loeve transform

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Actually, there is a much more powerful technique than inside-trace muting or optimum-weighted stacking to attenuate multiples. It is based on Karhunen-Loeve (K-L) transform [1] [2] [3]. The basic underlying concept is that one can decompose a two-dimensional data set, such as a stacked section in space-time coordinates, into a number of components — the so-called eigenimages, starting with the first eigenimage that contains the highest correlatable events, moving onto the next eigenimage with events with less degree of correlation, all the way to the last eigenimage that comprises the least correlatable components. Each eigenimage comprises the same number of traces as the original data set. Singular-value decomposition (Section F.3) is one way of decomposing a data set into its eigenimages.

So, how is this transform applicable to stacking and multiple suppression? If you apply NMO correction to CMP gathers using the primary velocities, then the primaries will be flattened and thus will have the highest correlatability from trace to trace in the gathers. Hence, these primaries will map into the first eigenimage of the K-L transform. By retaining only the first eigenimage and discarding the others, and stacking the traces in the first eigenimage, one can obtain a stacked trace that is virtually free of random and coherent noise such as multiples. Of course, in reality, there is always some noise component that creeps into the first eigenimage. This happens, for example, when there is very little moveout difference between primaries and multiples. An alternative strategy that is aimed at multiple suppression involves moveout correction using multiple velocities, rather than primary velocities.

Start with a modeled CMP gather as shown in Figure 6.1-12a that contains a primary (arriving at 0.2 s at zero offset) and the associated multiples. This gather also contains three additional primaries (at 0.4, 0.8 and 1.2 zero-offset times) that are weaker in amplitude compared to the multiples. The moveout difference between the multiples and primaries is less than 100 ms at the far offset.

Apply NMO correction using the multiple velocity, in this case constant 3000 m/s. Multiples are flattened, and primaries are overcorrected (Figure 6.1-12b). Then perform singular-value decomposition, which is the basis for K-L transform, and examine the first eigenimage (Figure 6.1-12c). Note that the highest correlatable events in this gather are those events with the moveout velocity of 3000 m/s — the primary at 0.2-s zero-offset time and the associated multiples. Subtract this eigenimage from the original NMO-corrected gather (Figure 6.1-12b) to get the gather that is a composite of all the eigenimages except the first one (Figure 6.1-13a). Note that multiples have been attenuated and the weak primaries have been retained. Because we did not reject the higher eigenimages associated with the least correlatable energy, including the random noise, this energy is present in the output gather in Figure 6.1-13a.

Finally, apply inverse NMO correction using the same multiple velocity function to obtain the gather after multiple attenuation (Figure 6.1-13b), which should be compared with the original gather without multiple attenuation (Figure 6.1-13c). Note that, the K-L transform, in principle, is a very powerful data decomposition technique that can be used to attenuate multiples.

In addition to multiple attenuation, the K-L transform also can be used to attenuate random noise by simply rejecting the corresponding eigenimages. For instance, a 60-trace CMP gather would be decomposed into 60 eigenimages. Those eigenimages with eigenvalues between, say 55 and 60, would contain the random noise to be rejected.

Figure 6.1-14a shows a CMP gather that contains strong water-bottom and peg-leg multiples. Following moveout correction using the water velocity, the water-bottom multiples are flattened, peg-legs are slightly overcorrected and primaries, which are visible at far offsets below 2 s, are significantly overcorrected (Figure 6.1-14b).

By the K-L transform, the gather is decomposed into its eigenimages. Figure 6.1-15 shows the reconstructed gather as in Figure 6.1-14b using only a subset of the eigenimages. The reconstructed gather using only the first eigenimage contains just a few of the strong moveout-corrected multiples. With the inclusion of additional eigenimages, the reconstructed gather is allowed to contain events with some moveout. By examining the series of reconstructed gathers with a subset of eigenimages, a band of eigenimages that corresponds to multiples can be determined.

Figure 6.1-14c shows such a reconstructed gather using the eigenimages that correspond to the first 20 eigenvalues. Subtract the gather in Figure 6.1-14c from that in Figure 6.1-14b to obtain the gather that presumably contains the primary reflections (Figure 6.1-14d). Following the application of inverse moveout correction (Figure 6.1-14e) the data are ready for velocity analysis after multiple attenuation. Figure 6.1-16 shows a portion of the CMP stack associated with the data in Figure 6.1-14 with and without multiple attenuation using the K-L transform. While this example demonstrates that the method can be successful in attenuating multiples associated with near-horizontal reflections, it also can be a robust technique in handling multiples associated with moderately complex reflections. Figure 6.1-17 shows a portion of a CMP stack that contains a strong primary at 1.2 s at the left-hand edge of the section. The K-L transform method has largely attenuated the water-bottom and peg-leg mutiples associated with the strong primary, and enhanced the primary events above it. Following the eigenimage decomposition, caution must be exercised in determining the number of eignevalues included in the reconstruction of CMP gathers (Figure 6.1-15).

As mentioned earlier, a flat event within a data window in time and space has the highest degree of correlation from trace to trace, and thus, maps into the first eigenimage. Doicin and Spitz [4] exploited this property in the frequency-space domain based on the earlier work by White [5] to better separate primaries from peg-leg multiples into different eigenimages. Consider a portion of a common-offset section in Figure 6.1-18a associated with a marine data set. The water bottom is nearly flat at approximately 150 ms. The common-offset section exhibits a primary reflection (K) associated with an erosional unconformity in the neighborhood of 3 s. The flat water bottom gives rise to a series of peg-leg multiples (M1 and M2) of the primary reflection (K), arriving at an equal time interval.

If we perform horizon flattening on the primary reflection (K), then the multiple reflections (M1 and M2) would also be flattened. The eigenimage decomposition of the data window that excludes the flattened primary but includes the flattened multiples maps these into the first eigenimage. Reject the first few eigenimages and reconstruct the data window from the remaining eigenimages without the peg-leg multiples. The final step involves unflattening of the data. This process is applied to each of the common-offset sections and the data are stacked. Note the absence of the peg-leg multiples on the stacked section shown in Figure 6.1-18b. This section lends itself to an improved image of the dipping events below the unconformity (Figure 6.1-18c).

Peg-leg multiples associated with a reflector below a dipping water bottom exhibit a complex traveltime behavior [6]. Figure 6.1-19 shows selected CMP gathers that include peg-leg multiples of a complex nature. A sketch of raypaths associated with peg-leg multiples is shown in Figure 6.1-20. For a horizontally layered earth model, the peg-leg raypath segment on the source end (the solid path) of the CMP ray-path and the peg-leg raypath segment on the receiver end (the dotted path) of the CMP raypath give rise to coincident arrival times on the CMP gather (Figure 6.1-20a). When there are dipping reflectors along the raypaths of the peg-leg multiples, the peg-leg raypath segment on the source end of the CMP raypath and the peg-leg raypath segment on the receiver end of the CMP raypath give rise to split peg-leg multiple arrivals (Figure 6.1-20b). Also note that, while the minimum arrival time of the primary reflection is at near-offset trace, the minimum arrival time of the peg-leg multiple reflections is at some nonzero-offset trace. The split peg-leg multiples can be extremely troublesome when interpreting data recorded over a continental slope.

## References

1. Jones and Levy, 1987, Jones, I. F. and Levy, S., 1987, Signal-to-noise ratio enhancement in multichannel seismic data via the Karhunen-Loeve transform: Geophys. Prosp., 35, 12–32.
2. Ulrych et al., 1988, Ulrych, T. J., Freire, S. L. and Siston, P., 1988, Eigenimage processing of seismic sections: 58th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1261–1265.
3. Al-Yahya, 1991, Al-Yahya, K., 1991, Application of the partial Karhunen-Loeve transform to suppress random noise in seismic sections: Geophys. Prosp., 39, 77–93.
4. Doicin and Spitz (1991), Doicin, D. and Spitz, S., 1991, Multichannel extraction of water-bottom peg-legs pertaining to gihg-amplitude reflection: 61st Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1439–1442.
5. White (1984), White, R. E., 1984, Signal and noise estimation from seismic reflection data using spectral coherence methods: Proc. IEEE, 72, 1340–1356.
6. Levin and Shah, 1977, Levin, F. K. and Shah, P. M., 1977, Peg-leg multiples and dipping reflectors: Geophysics, 42, 957–981.