Source signature estimation based on the removal of first order multiplesLuc T. Ikelle*, Graham Roberts, and Arthur B. Weglein, Schlumberger Cambridge Research

SP5.7

SUMMARY

The estimation of the source signature is often one of thenecessary first steps in the processing of seismic reflectiondata, especially if the processing chain includes prestackmultiple removal. In this work, a new source estimationmethod based on prestack data is presented. It consistsof finding the source signature that permits the removalof events due to the first order free-surface reflections (i.e.first order multiples). The method exploits the formula-tion of the relationship between the free-surface reflectionsand the source signature as an inverse scattering Born se-ries. In this formulation, the order of the scattering seriescoincides with that of the free-surface reflections, and theseries is constructed exclusively with seismic data, the wa-ter velocity and source signature, and without requiring anyknowledge of the subsurface.

By restricting the problem to first order free-surface re-flections, we have rendered the relationship between free-surface reflections and source signature linear, which alsocorresponds to a truncation of the inverse scattering Bornseries to its first two terms. Thus, the source signatureestimation can be formulated as a linear inverse problem.Assuming that the removal of first order free-surface eventsproduces a significant reduction in the energy of the data,we posed the inverse problem as finding the source signa-ture which minimizes this energy. This optimization leadsto a stable iterative solution. The iterations are needed tocorrect for the truncation effects. The preliminary resultswith real and synthetic data are quite satisfactory.

SOURCE ESTIMATION AND FREE-SURFACEMULTIPLE REMOVAL

Free-surface multiple removal

The estimation of the source signature is one of the classicproblems in exploration seismology. Although linear pro-cessing methods can derive benefits from knowledge of thesource signature, its detailed knowledge is an essential pre-requisite for non-linear methods such as the surface multipleremoval procedure.

The approach to wavelet inversion described here consistsof finding the source signature which optimally removes the

first order multiples within the framework of an inverse scat-tering series. It exploits the relationship between the free-surface reflections and the source signature formulated as ascattering Born series [Carvalho et al. (1991)]. Each termof this series is constructed using exclusively the seismic re-flection data, the velocity of water and the source signature.No knowledge of the subsurface is required.

The scattering Bornples can be written

series for removing free-surface

(1)

where is the Fourier transform of the seismicdata with respect to receiver and source locations and time and are the horizontal wavenumbers forreceiver and source, respectively, and w is the frequency. denotes data without free-surface multiples and A denotesthe inverse of the Fourier transform of the source signature.

etc. are given by

= c

, (2)

(3)The constant c is the velocity of water and k is a generic hor-izontal wavenumber. The importance of the obliquity fac-tor, cos (plane wave incident angle for free surface mul-tiple removal has recently been demonstrated by Dragoset(1993), but it's sometimes omitted in other formulations.The scattering series described in equation (1) shows thatthe removal of the free-surface multiples involves only theseismic data, and the inverse source, A. The first termof the scattering series, is the deghosted data; the sec-ond term, removes first order free-surface multiples;the next term, removes second order free-surface mul-tiples, and so on. Verschuur et al. (1992) and Carvalhoand Weglein (1994) have used two different approaches forestimating the wavelet from the entire series. Their meth-ods are based on the idea that data without multiples hasless energy than data with multiples, a non-linear problemthat requires a numerical rather than an analytic solution.

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Source estimation/multiple removal 2

In contrast, the procedure presented here is based on theconcept that data without first order surface mutiples hasless energy than data with those multiples and results ina linear problem with an analytic solution. Iterations arealso analytic.

In our approach to source estimation, we truncate the seriesto its first two terms,

(4)which essentially corresponds to the removal of the first or-der multiples. The theory predicts that if we find a sourcesignature which permits the removal of the first order mul-tiples, such source signature will also remove higher ordermultiples when it is used with higher order terms of theseries. Furthermore, we have rendered the relationship be-tween and A linear. Thus the source signature estima-tion can be formulated as a linear inverse problem.

Source estimation

Assuming that the removal of the first order free-surfaceevents produces a lowering in energy of the data, we posedthe inverse problem as finding the best source signaturewhich minimizes this energy. The best source in the leastsquares sense is defined as the A which minimizes

(5)where denotes the weighted norm.

If we neglect truncation errors in the forward problem asgiven in equation (4), the minimization problem in equation(5) gives the following analytical solution

(6)

where the asterisk denotes complex conjugate. The opti-mization algebra that leads to this solution is similar toIkelle et al. (1986). The weighting function, W, describesthe errors in data and the a priori information on the source.The application of equation (4), using the inverse source permits a significant reduction of first order multiple energy.Iterations are performed to correct allowing the removalof the remaining first order multiple energy. For that pur-pose, we set equal to the result in equation (4), and thenreapplied equation (6) to find the correction to the source.This process is repeated until two successive corrections aresufficiently close.

SYNTHETIC DATA EXAMPLE

In this section, we present a synthetic example to illustratethe method. Our model consists of two homogeneous hori-zontal layers overlying a homogeneous half space. The inter-faces are at 120 and 285 meters. The layer and half space ve-locities are 1500 m/s, 2000 m/s and 2500 m/s, respectively.A synthetic shot point gather was generated, consisting of83 receivers with a spacing of 12.5 metres.

Figure 1 displays the synthetic data. The data consists oftwo primaries and several free-surface multiples. In this ex-ample, the primaries and multiples interfere. In particular,notice that at near offsets the second primary, and thefirst order multiple of the first primary, interfere destruc-tively, but as offset increases the extent of the interferencedecreases due to the difference in moveout between the pri-mary and multiple. This means that, in this example, atnear offsets energy increases when first order multiples areremoved, contrary to the energy criteria required for op-timization. However, as offset increases the energy of datawith multiples becomes greater than the energy of data withprimaries only. Hence, for this type of situation we must en-sure that the range of offsets is adequate to allow the properapplication of this method.

Figure 2 shows the results of the inversion for the sourcesignature for the first and fifth iterations. The result isquite satisfactory when compared to the actual source. Thephase is well estimated after only one iteration with fur-ther iterations required to improve the amplitude. Figure3 illustrates the effect of multiple removal using the sourceafter one iteration only. We see that the multiple of hasbeen suppressed but not totally removed. The result after5 iterations is shown in Figure 4; all multiples have beensuccessfully removed.

REAL DATA EXAMPLE

Let us now look at an application of our source estimation toreal data. The data is a 2D marine seismic line acquired byTotal which has been used in previous multiple suppressionstudies [e.g., Jugla et al. (1994)]. The preprocessing hereconsisted of despiking, muting the direct wave and a 3D to2D amplitude correction.Figure 5 shows the source wavelet after one and five it-erations. Notice that the iteration process has essentiallycorrected only the amplitude of the wavelet. As in the syn-thetic example discussed earlier, the phase of the wavelet islargely unchanged through the iteration process. We havealso observed, based on the 185 shot gathers used in thissource estimation, that source directivity and variations in

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3 Source estimation/multiple removal

the source wavelet from shot to shot are not significant inthis case. However, when these effects are significant, theycan be accomodated by the solution for the inverse sourcegiven in equation (6).

Ikelle, L.T., Diet, J.P. and Tarantola, A., 1986, Linearizedinversion of multioffset seismic reflection data in the domain. Geophysics, 51, 1266-1276.

Figure 6 shows shot gathers before and after multiple sup-pression using the source wavelet after five iterations alongwith the estimated multiples. The efficacy of the multi-ple removal method is demonstrated by the lower multipleenergy between 0.6 and 1.5 seconds and the removal of themultiple from the interfering event at 1.5 seconds.

Jugla, F., Julien, P., de Groot, P.F. and Verschuur, E.,1994, A real case comparison between surface-related multi-ple elimination and other wave equation-based techniques.Presented at the 64th SEG meeting, Los Angeles.

Verschuur, D.J., Berkhout, A.J. and Wapenaar, C.P.A.,1992, Adaptive surface-related multiple elimination. Geo-physics, 57, 1166-1177.

Spikes were present in the data and some were removed.The other spikes could not be easily removed without al-tering the primaries. Therefore, these latter spikes are re-tained. They are interpreted as primaries and the methodproduces their demultiple operators. This manifests as aseries of diffractions on the demultiple gather and the esti-mated multiples.

CONCLUSIONS

For successful prestack free-surface multiple removal, a knowl-edge of the source signature is required. By analogy to thevelocity-migration method for estimating velocity using mi-gration, we have exploited this requirement to estimate thesource which permits the free-surface multiple removal. Byrestricting the problem to first-order multiples, we have ren-dered it linear and analytic in solution. Each iteration isalso analytic. Results with real and synthetic data showedthat the main characteristic of the source is recovered at thefirst iteration. Approximately five iterations are needed forsuccessful multiple removal.

The method results in a good estimate of the wavelet formultiple attenuation and other applications.

Figure 1. Shot record corresponding to a subsurfacemade of two homogeneous horizontal layers and a homo-geneous half space. The data has only two primaries, and but several multiples. The first order multiple of

interferes with primary,

A C K N O W L E D G E M E N T S

The authors acknowledge the Norwegian Petroleum Direc-torate and Total CFP for permission to publish the data.

R E F E R E N C E S

Carvalho, P.M., Weglein, A.B. and Stolt, R.H., 1991, Exam-ples of a Nonlinear Inversion Method Based on the T Matrixof Scattering Theory: Application to Multiple Suppression.Presented at the 61st SEG meeting, Houston.Carvalho, P.M., Weglein, A.B. 1994, Wavelet Estimation forSurface Multiple Attenuation using a Simulated AnnealingAlgorithm. Presented at 64th SEG meeting, Los Angeles.

Dragoset, W. and MacKay S., 1993, Surface Multiple At-tenuation and Subsalt Imaging. Presented at the 63rd SEG

Figure 2. Actual (solid) and estimated source waveletafter 1 (dashed) and 5 (dotted) iterations.

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Source es t imat ion /mu l t ip le remova l 4

Figure 3. Result of multiple removal on the shot record Figure 4. Result of multiple removal on the shot recordof Figure 1 using the source signature obtained after 1 of Figure 1 using source signature obtained after 5 itera-iteration. tions.

Figure 5. Estimated source wavelet after 1 (dashed)and 5 (solid) iterations from real data.

Figure 6. From left to right: shot gather before multiple elimination, aftermultiple elimination and the predicted multiples.

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