A new inverse scattering series internal-multiple

A new Inverse Scattering Series (ISS) internal-multiple-attenuation algorithm that predicts the ac-curate time and approximate amplitude of the first-order internal multiples and addresses spuriousevents: Analysis and Tests in 2DChao Ma ∗ and Arthur B. Weglein, M-OSRP/Physics Dept./University of Houston

SUMMARY

The ISS internal-multiple-attenuation algorithm assumes thatthe free-surface multiples have been removed from the inputof this algorithm, and the input consists of only primaries andinternal multiples. The algorithm selects three events by a“longer-shorter-longer” relationship in the vertical-travel-timedomain and the primaries selected in that procedure predict theaccurate time and approximate amplitude of all first-order in-ternal multiples without any subsurface information (Wegleinet al., 2003). However, the input data contain both primariesand internal multiples. When internal multiples themselvesare selected in that procedure, two different types of eventswill be produced. The first type is higher-order internal mul-tiples (e.g., second-order internal multiples (Zhang and Shaw,2010)), and these predicted higher-order internal multiples willcooperatively assist and benefit the attenuating of higher-orderinternal multiples in the data. The second type is spuriousevents (events that do not exist in the data). For the sec-ond type of events, Weglein et al. (2011), Liang et al. (2013)and Ma and Weglein (2014) show that the spurious events arefully anticipated by the ISS, and specific higher-order termsfrom ISS will precisely address that spurious-event issue. Theinclusion of higher-order terms provides a new ISS internal-multiple-attenuation algorithm that does not generate the spu-rious events and, at the same time, retains the strength of theoriginal algorithm. That original contribution (i.e., the afore-mentioned new algorithm) was for a one dimensional subsur-face. In this paper, we extend the previous work on addressingthe spurious events to a multi-D case and show more realisticsynthetic test results in 2D. Those tests exemplify the relevantand practical benefit provided by this new internal-multiple al-gorithm.

INTRODUCTION

The inverse scattering series (ISS) communicates that it is pos-sible to achieve all seismic data processing objectives directlyand without subsurface information. The current ISS internal-multiple-attenuation algorithm was first developed by Araujoet al. (1994) and Weglein et al. (1997). The unique strength (al-ways present independent of the circumstances and complex-ity of the geology and the play) of the ISS internal-multiple-attenuation algorithm is that this algorithm is able to predictinternal multiples without any subsurface information. Hence,the ISS internal-multiple-attenuation algorithm is often calledupon in the cases in which the multiple-removal is a chal-lenging problem and it is difficult to find the subsurface in-formation for other multiple-suppression methods to be effec-tive. The tests on ISS internal-multiple-attenuation algorithmhave shown promising results and unique value compared

with other multiple-suppression methods (e.g., K.Maston et al.(1999); Fu et al. (2010); Hsu et al. (2010); Ferreira (2011);Terenghi et al. (2011); Luo et al. (2011); Weglein et al. (2011);Kelamis et al. (2013)).

Early analysis of the current ISS internal-multiple-attenuationalgorithm focused on selecting primaries in the input to pre-dict internal multiples. However, the input data contain bothprimaries and internal multiples and all events in the data willbe selected. Internal multiples selected in this algorithm cangenerate spurious events under the circumstances where threeor more reflectors are involved in the data being processed asshown by Weglein et al. (2011), Liang et al. (2013) and Maand Weglein (2014).

The work of Ma and Weglein (2014) also demonstrates thatspurious-event issue is serious and significant when there aretens, hundreds (or even thousands) of internal-multiple gen-erators (e.g., Middle East and North Sea), and addressing thespurious events under those circumstances is essential by ap-plying a new ISS algorithm that addresses the spurious-eventprediction and retains the strength of the original algorithm si-multaneously. In this paper, we extend the analysis of address-ing the spurious events to a multi-D case, and also provide amore realistic numerical test in 2D.

THE CURRENT ISS INTERNAL-MULTIPLE-ATTENUATION ALGORITHM

The current ISS internal-multiple-attenuation algorithm startswith the input data, D(kg,ks,ω), in 2D case, which are theFourier transform of the deghosted prestack data, and with thewavelet deconvolved and direct wave and free-surface multi-ples removed. The second term, D3(kg,ks,ω), is the attenuatorof the first-order internal multiples. In a 2D earth, D3(kg,ks,ω)is obtained from b3(kg,ks,ω) = −2iqsD3(kg,ks,ω), whereb3(kg,ks,ω) is (Weglein et al., 2003)

b3(kg,ks,qs +qs) =1

(2π)2

∫ ∞

−∞dk1

∫ ∞

−∞dk2e−iq1(zg−zs)eiq2(zg−zs)

×∫ ∞

−∞dz1b1(kg,k1,z1)ei(qg+q1)z1

×∫ z1−ε

−∞dz2b1(k1,k2,z2)e−i(q1+q2)z2

×∫ ∞

z2+εdz3b1(k2,ks,z3)ei(q2+qs)z3 , (1)

where ks and kg are the horizontal wavenumbers for thesource and receiver coordinates, respectively; qg and qs arethe vertical source and receiver wavenumbers defined by qi =

sgn(ω)

√ω2

c20− k2

i for i ∈ {g,s} (ω is the temporal freqency);

SEG New Orleans Annual Meeting Page 4402

DOI http://dx.doi.org/10.1190/segam2015-5834382.1© 2015 SEG

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zs and zg are source and receiver depths; and z j (i ∈ {1,2,3})represents pseudo-depth by using a reference velocity migra-tion. The quantity b1(kg,ks,z) corresponds to an uncollapsedmigration (Weglein et al., 1997) of effective plane-wave inci-dent data.

The data with their first-order internal multiple attenuated are

D(kg,ks,ω)+D3(kg,ks,ω). (2)

For a 1-D earth and a normal incident plane wave, equation 1reduces to

b3(k) =∫ ∞

−∞dz1eikz1 b1(z1)

∫ z1−ε

−∞dz2e−ikz2 b1(z2)

×∫ ∞

z2+εdz3eikz3 b1(z3). (3)

The deghosted data, D(t), for an incident plane wave, sat-isfy D(ω) = b1(

2ωc0), D(ω) is the temporal Fourier transform

of D(t), b1(z) =∫ ∞−∞ eikzb1(k)dk, and k = 2ω

c0is the vertical

wavenumber.

Equation 2 then reduces to

D(t)+D3(t), (4)

where D3(t) is Inverse Fourier transform of D3(ω), andD3(ω) = b3(

2ωc0), where k = 2ω

c0.

Figure 1 illustrates the idea behind using equation 1 or equa-tion 3 to predict the first-order internal multiple by selectingprimaries (events that experience only one upward reflection)in the data as subevents, and combining different subeventsthat satisfy the “ longer(A)-shorter(B)-longer(C) ”relationshipin vertical-travel-time domain (or equivalently, “ lower(A)-higher(B)-lower(C) ” relationship in pseudo-depth domain(Nita and Weglein, 2007)).

Figure 1: An internal multiple (dashed line) constructed by thelower-higher-lower pattern of three primary subevents (solidline). Figure adapted from Weglein et al. (2003)

THE ORIGIN OF SPURIOUS EVENTS AND ITS RESO-LUTION IN 1D

The work of Araujo et al. (1994) and Weglein et al. (1997) fo-cuses on the analysis of the prediction of first-order internalmultiples (i.e., equation 1) by using primaries in the data as

Figure 2a: In a two-reflector example, a “Primary – Primary –Internal multiple” combination predicts a second-order inter-nal multiple.

Figure 2b: In a three-reflector example, a “Primary – Internalmultiple – Primary” combination predicts a spurious event.

subevents (see Figure 1). However, data consist of both pri-maries and internal multiples. Zhang and Shaw (2010) showthat higher-order internal multiples will be predicted by b3when internal multiples themselves are selected as a subeventsin a two-interface case. For example, in Figure 2a, a second-order internal multiples will be predicted when a first-orderinternal multiple is selected as a subevent.

In addition, the situation is considerably more complicatedwhen the data from three or more reflectors are considered. Forinstance, spurious events can also be generated when an inter-nal multiple is selected as a subevent in a three-reflector ex-ample, as shown in Figure 2b. However, these spurious eventsare also entirely anticipated by the inverse scattering series andthere are terms in the series that can exactly address those falseevent prediction.

After identifying all the terms in the series that address thespurious events, a new ISS internal-multiple-attenuation algo-rithm is provided by including those terms. The new algorithmboils down as follows,

D(t)+DNew3 (t), (5)

where DNew3 (t) is the Inverse Fourier transform of DNew

3 (ω),DNew

3 (ω) = bNew3 (k = 2ω

c0), and bNew

3 (k) is obtained from

bNew3 (k) =

∫ ∞

−∞dz1eikz1(b1(z1)+b3(z1))

×∫ z1−ε

−∞dz2e−ikz2(b1(z2)+b3(z2))

×∫ ∞

z2+εdz3eikz3(b1(z3)+b3(z3)). (6)

The new algorithm will address the spurious events by reduc-ing the internal multiples using (b1(z)+b3(z)) as the new in-put.



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Figure 3a: In a 2D case, the workflow of the current ISSinternal-multiple-attenuation algorithm. Step 1 is the uncol-lapsed Stolt migration, step 2 is the prediction from the atten-uator of the first-order internal multiples, step 3 transforms theprediction back to space-time domain.

A NEW ISS INTERNAL-MULTIPLE-ATTENUATIONALGORITHM THAT PREDICTS THE ACCURATETIME AND APPROXIMATE AMPLITUDE OF THEFIRST-ORDER INTERNAL MULTIPLE AND AD-DRESSES SPURIOUS EVENTS

By understanding the generation of the spurious events and itsresolution in 1D case, we provide a new multi-D ISS internal-multiple-attenuation algorithm (Figure 3b) that addresses thespurious events and preserves the strength of the current algo-rithm. For the purpose of comparison, we show the current ISSinternal multiple attenuation algorithm in Figure 3a.

We test the new algorithm using a synthetic 2D data set. Figure4 shows a three-reflector model used to generate the 2D syn-thetic data set by finite-difference method. The data consistof 251 shots × 251 receivers, with both shot- and receiver-interval 25 m, each trace has 500 samples with a total duration4s. The internal multiples will be strong because of the bigimpedance contrast between layers. Figure 5 and 6 show oneshot and trace comparison between the test data and the pre-diction results with (Figure 5b and 6b) and without (Figure5a and 6a) addressing the spurious-event prediction. In Fig-ure 5, black and red arrows point to the primaries and internalmultiples, respectively. The numbers in the subscript indicatethe reflectors where the reflection happens. The blue arrows inFigure 5 indicate the places where the prediction result withthe addressing of spurious events better matches the test datathan that without addressing the spurious events (see black ar-rows in Figure 6 for details). Compared with Figure 5a (or 6a),the prediction of higher-order internal multiples (e.g., I21212)in Figure 5b (or 6b) gets reduced (see red arrows in Figure6 for details) because of the reduced internal multiple in theinput data.

Figure 3b: In a 2D case, the workflow of the new ISS internal-multiple-attenuation algorithm with the addressing of the spu-rious events. The steps are same as in Figure 3a, but using anew input. D3(xg,xs; t) is the output from the current algorithmin Figure 3a.

Figure 4: Synthetic velocity and density model used to gener-ate the test data in this section (courtesy of WesternGeco). Theaverage dip of the walls of the trench featuring in the centerof the model is approximately 20 degree (Figure adapted fromTerenghi and Weglein (2011)).

CONCLUSIONS

In this paper, we analyze, develop and test a new multi-D ISSinternal-multiple-attenuation algorithm that anticipates and re-moves the spurious events that are generated by the current ISSinternal-multiple-attenuation algorithm. The numerical test ona synthetic 2D data set in this paper shows the added value ofapplying the new algorithm to address the spurious events thatcan be significant many real-world applications. This issuewill arise when many reflectors generate the multiples in com-plex on-shore and off-shore plays, and can be serious impedi-ment to interpretation and making effective drilling decisions.The new algorithm in this paper addresses that issue.

ACKNOWLEDGMENTS

We are grateful to all M-OSRP sponsors for their encourage-ment and support of this research. WesternGeco is gratefullyacknowledged for permission to use their synthetic data.



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Figure 5a: A shot comparison between the test data (left part)and the ISS internal multiple prediction (right part) withoutaddressing the spurious events.

Figure 5b: A shot comparison between the test data (left part)and the ISS internal multiple prediction (right part) with ad-dressing the spurious events.

Figure 6a: A trace comparison (from 1.4s to 2.6s) betweenthe test data (red line) and the ISS internal multiple prediction(blue line) without addressing the spurious events.

Figure 6b: A trace comparison (from 1.4s to 2.6s) betweenthe test data (red line) and the ISS internal multiple prediction(blue line) with addressing the spurious events.



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EDITED REFERENCES Note: This reference list is a copyedited version of the reference list submitted by the author. Reference lists for the 2015 SEG Technical Program Expanded Abstracts have been copyedited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

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Ferreira, A., 2011, Internal multiple removal in offshore Brazil seismic data using the inverse-scattering series: Ph.D. thesis, University of Houston.

Fu, Q., Y. Luo, P. G. Kelamis, S. Huo, G. Sindi, S.-Y. Hsu, and A. Weglein, 2010, The inverse-scattering series approach toward the elimination of land internal multiples: 80th Annual International Meeting, SEG, Expanded Abstracts, 29, 3456–3461.

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A new inverse scattering series internal-multiple