The first Inverse-Scattering-Series internal multiple elimination method for a multi-dimensional subsurfaceYanglei Zou, Chao Ma, Arthur B. Weglein, M-OSRP/Physics Dept./University of Houston

SUMMARY

The ISS (Inverse-Scattering-Series) internal-multiple attenua-tion algorithm (Araujo et al. (1994),Weglein et al. (1997) andWeglein et al. (2003)) is the most effective algorithm todayfor internal multiple removal. It is the only multi-dimensionalmethod that can predict the correct time and approximate am-plitude for all internal multiples at once, without any subsur-face information. When combined with an energy minimiza-tion adaptive subtraction, the ISS internal-multiple attenuationalgorithm can effectively eliminate internal multiples when theprimaries and internal multiples are separated. However, undermany offshore and onshore circumstances where internal mul-tiples are often proximal to or interfering with primaries, thecriteria of energy minimization adaptive subtraction can fail(e.g., the energy can increase when a multiple is removed froma destructively interfering primary and multiple). Therefore,Weglein (2014) proposed a three-pronged strategy for provid-ing an effective response to removing internal multiples with-out damaging interfering primaries. Currently, there is no ca-pability available in the petroleum industry that addresses thattype of serious and frequently occurring challenge. A majorcomponent of the strategy is to develop an internal-multipleelimination algorithm that can predict both the correct am-plitude and correct time for all internal multiples. The ini-tial idea to achieve an elimination algorithm is developed byWeglein and Matson (1998) by removing attenuation factors(the difference between the predicted internal multiples andtrue internal multiples) using reflection data. There are earlydiscussions in Ramırez (2007). Based on the ISS attenuationalgorithm and the initial idea for elimination, Herrera and We-glein (2012) formulated an ISS algorithm for a normal incidentwave on a 1D earth, that eliminate first-order internal multi-ples generated by the shallowest reflector and further attenu-ates first-order internal multiples from deeper reflectors. Zouand Weglein (2014) then advanced and extended these initialcontributions for the pre-stack and for all first order internalmultiples generated at all reflectors. In this paper, we furtherextend the 1-D elimination algorithm and provide the first ISSmulti-dimensional elimination method for all first order inter-nal multiples.

INTRODUCTION

The ISS (Inverse-Scattering-Series) allows all seismic process-ing objectives, e.g., free-surface-multiple removal and internal-multiple removal to be achieved directly in terms of data, with-out any need for or estimation of the earth’s properties. TheISS internal-multiple attenuation algorithm is the only methodtoday that can predict the correct time and approximate andwell-understood amplitude for all first-order internal multiplesgenerated from all reflectors, at once, without any subsurface

information. If the multiple to be removed is isolated fromother events, then the energy minimization adaptive subtrac-tion can fill the gap between the attenuation algorithm ampli-tude prediction and the internal multiples plus, e.g., all prepro-cessing factors that are outside the assumed physics of the sub-surface and acquisition. However primary and multiple eventscan often interfere with each other in both on-shore and off-shore seismic data. In these cases, the criteria of energy mini-mization adaptive subtraction may fail and completely remov-ing internal multiples becomes more challenging and beyondthe current capability of the petroleum industry.

For dealing with this challenging problem, Weglein (2014)proposed a three-pronged strategy including

1. Develop the ISS prerequisites for predicting the refer-ence wave field and to produce de-ghosted data.

2. Develop internal-multiple elimination algorithms fromISS.

3. Develop a replacement for the energy-minimization cri-teria for adaptive subtraction.

To achieve the second part of the strategy, that is, to upgradethe ISS internal-multiple attenuation algorithm to eliminationalgorithm, the strengths and limitations of the ISS internal-multiple attenuation algorithm are noted and reviewed. TheISS internal-multiple attenuation algorithm always attenuatesall internal multiples from all reflectors at once, automaticallyand without any subsurface information. That unique strengthalways present and is independent of the circumstances andcomplexity of the geology and the play. However, there aretwo well-understood limitations of this ISS internal-multipleattenuation algorithm

1. It may generate spurious events due to internal multi-ples treated as sub-events.

2. It is an attenuation algorithm not an elimination algo-rithm.

The first item is a shortcoming of the leading order term (theterm used to derive the current attenuation algorithm), whentaken in isolation, but is not an issue for the entire ISS internal-multiple removal capability. It is anticipated by the ISS andhigher order ISS internal multiple terms exist to precisely re-move that issue of spurious events prediction. When taken to-gether with the higher order terms, the ISS internal multipleremoval algorithm no longer experiences spurious events pre-diction. Ma et al. (2012) , H. Liang and Weglein (2012) andMa and Weglein (2014) provided those higher order terms forspurious events removal.

In a similar way, there are higher order ISS internal multipleterms that provide the elimination of internal multiples when

Internal Multiple Removal

taken together with the leading order attenuation term. The ini-tial idea is provided by Weglein and Matson (1998) in whichthe attenuation factor, which is a collection of extra transmis-sion coefficients and is the difference between attenuation andelimination, is systematically studied. Later there are furthurdiscussions in Ramırez (2007). Several extensions are pro-posed based on the initial idea. Herrera and Weglein (2012)proposed an algorithm for internal multiple elimination for allfirst order internal multiples generated at the first reflector.Benefited from the previous work, Zou and Weglein (2014)proposed an new algorithm that can eliminate all first orderinternal multiples for all reflectors for a 1D earth. In this pa-per, we further extend the previous elimination algorithm andprovide the first ISS multi-dimensional elimination method forall first order internal multiples. The new elimination algo-rithm retains the benefits of the attenuation algorithm, includ-ing not requiring any subsurface information and unlike strip-ping methods, removes all first-order internal multiples fromall subsurface reflectors at once.

THE ISS INTERNAL-MULTIPLE ATTENUATION AL-GORITHM AND THE INITIAL IDEA FOR INTERNALMULTIPLE ELIMINATION

The ISS internal-multiple attenuation algorithm is first givenby Araujo et al. (1994) and Weglein et al. (1997). The 1Dnormal incidence version of the algorithm is presented as fol-lows (The 2D version is given in Araujo et al. (1994),Wegleinet al. (1997) and Weglein et al. (2003) and the 3D version is astraightforward extension.),

b3(k) =∫

∞

−∞

dzeikzb1(z)∫ z−ε2

−∞

dz′e−ikz′b1(z′)

×∫

∞

z′+ε1

dz′′eikz′′b1(z′′). (1)

Where b1(z) is the constant velocity Stolt migration of the dataof a 1D normal incidence spike plane wave. ε1 and ε2 aretwo small positive numbers introduced to avoid self interac-tions. bIM

3 (k) is the predicted internal multiples in the verticalwavenumber domain. This algorithm can predict the correcttime and approximate amplitude of all first-order internal mul-tiples at once without any subsurface information.

The procedure of predicting a first-order internal multiple gen-erated at the shallowest reflector is shown in figure 1. TheISS internal-multiple attenuation algorithm automatically usesthree primaries in the data to predict a first-order internal mul-tiple. (Note that this algorithm is model type independent andit takes account all possible combinations of primaries that canpredict internal multiples.) From this figure we can see, ev-ery sub event on the left hand side experiences several phe-nomena making its way down to the earth then back to the re-ceiver. When compared with the internal multiple on the righthand side, the events on the left hand side have extra trans-mission coefficients as shown in red. Multiplying all thoseextra transmission coefficients, we get the AF (attenuation fac-tor) - T01T10 for this first-order internal multiple generated at

the shallowest reflector. And all first-order internal multiplesgenerated at the shallowest reflector have the same attenuationfactor.

Figure 2 shows the procedure of predicting a first-order inter-nal multiple generated at the next shallowest reflector. In thisexample, the attenuation factor is (T01T10)

2(T12T21).

Figure 1: an example of the attenuation factor of a first-orderinternal multiple generated at the shallowest reflector, noticethat all red terms are extra transmission coefficients

Figure 2: an example of the attenuation factor of a first-orderinternal multiple generated at the next shallowest reflector, no-tice that all red terms are extra transmission coefficients

The attenuation factor for predicting a multiple generated bythe ith reflector, AFj , is given by the following:

AFj =

T0,1T1,0 ( f or j = 1)∏ j−1

i=1(T 2

i−1,iT2

i,i−1)Tj, j−1Tj−1, j ( f or 1 < j < J)

(2)

The subscript j represents the generating reflector, and J is thetotal number of interfaces in the model. The interfaces arenumbered starting with the shallowest location. The attenua-tion factor is a collection of extra transmission coefficients andis the difference between attenuation and elimination. Wegleinand Matson (1998) studied the attenuation factor and providethe initial idea and algorithm to remove the attenuation factorby reflection data to achieve the elimination.

As discussed in Weglein and Matson (1998), the attenuationalgorithm prediction contains the attenuation factor and in or-der to develop an elimination algorithm, we should remove theattenuation factor. However, the attenuation factor is expressedusing transmission coefficients. Since the data contains reflec-tion coefficients, the idea is to use reflection coefficients to rep-resent the transmission coefficient such that we can remove theattenuation factor by the data (which contains reflection coef-ficients) without any subsurface information. For example, toremove AF1 in the prediction, we have

1AF1

=1

T0,1T1,0=

11−R2

1= 1+R2

1 +R41 + ... (3)

Internal Multiple Removal

where the first term 1 corresponds to the attenuation algorithm,the term R2

1 corresponds to the first higher order term towardselimination and so on. A detailed discussion can be found inWeglein and Matson (1998) and Ramırez (2007).

PREVIOUS WORK: WILBERTH’S AND YANGLEI’S EX-TENSIONS

Based on the ISS internal-multiple attenuation algorithm andthe initial idea for elimination, Herrera and Weglein (2012)formulated an algorithm for internal multiple elimination forall first order internal multiples generated at the first reflector.The first term in this elimination algorithm is the current atten-uator (equation (1)), which corresponds to the first term 1 inequation (3). The second term in this elimination algorithm isshown as follows (The complete algorithm is given in Herreraand Weglein (2012)),

+∞∫−∞

dzb1(z)e2ikzz−ε∫−∞

dz′F(z′)e−2ikz′+∞∫

z′+ε

dz′′b1(z′′)e2ikz′′ (4)

Where F(z) is

F(z) =

+∞∫−∞

d(2k)e−2ikz+∞∫−∞

dz′b1(z′)e2ikz′

×z′+ε∫

z′−ε

dz′′b1(z′′)e−2ikz′′z′′+ε∫

z′′−ε

dz′′′b1(z′′′)e2ikz′′′ (5)

The purpose of F(z) in the middle integrand is to provide R2

term shown in equation (3) in order to remove the attenuationfactor.

Benefited from the initial idea and Wilberth’s work, Zou andWeglein (2014) formulated an elimination algorithm that caneliminate all first order internal multiples for all reflectors fora 1D earth. Below shows the elimination algorithm, whereb1(k,z) is the water speed uncollapsed Stolt migration of thedata; bE(k,2q) is the elimination algorithm prediction in wavenum-ber domain; F [b1(k,z)] and g(k,z) are two intermediate func-tions.

bE(k,2q) =∫

∞

−∞

dze2iqzb1(k,z)∫ z−ε1

−∞

dz′e−2iqz′F [b1(k,z′)]

×∫

∞

z′+ε2

dz′′e2iqz′′b1(k,z′′) (6)

F [b1(k,z)] =1

2π

∫∞

−∞

∫∞

−∞

dz′dq′

× e−iq′zeiq′z′b1(k,z′)

[1−∫ z′−ε

−∞dz′′b1(k,z′′)eiq′z′′

∫ z′′+ε

z′′−εdz′′′g(k,z′′′)e−iq′z′′′ ]2

× 1

1−|∫ z′+ε

z′−εdz′′g(k,z′′)eiq′z′′ |2

(7)

g(k,z) =1

2π

∫∞

−∞

∫∞

−∞

dz′dq′

× e−iq′zeiq′z′b1(k,z′)

1−∫ z′−ε

−∞dz′′b1(k,z′′)eiq′z′′

∫ z′′+ε

z′′−εdz′′′g(k,z′′′)e−iq′z′′′

(8)

All the ISS internal multiple removal algorithms predict in-ternal multiples at once, without requiring any subsurface in-formation. For first -order internal-multiples generated at theshallowest reflector, both extensions are able to eliminate. Forfirst -order internal-multiples generated at the deeper reflec-tors, the Wilberth’s extension is still an attenuator and is ex-pected to predict better amplitude than the current attenuatorwhile the Yanglei’s extension is an eliminator.

THE FIRST INVERSE-SCATTERING-SERIES INTER-NAL MULTIPLE ELIMINATION METHOD FOR A MULTI-DIMENSIONAL SUBSURFACE

The Inverse-Scattering-Series contains an internal-multiple elim-ination sub-series. Since the internal multiple attenuation al-gorithm is capable to predict the correct time and approxi-mate amplitude for all internal multiples, if we can isolate allterms that can predict the same time as the attenuation algo-rithm by the initial elimination idea (removing the attenuationfactor by the reflection data) in the Inverse-Scattering-Series,then adding all these terms together will give us an elimina-tion algorithm. Since the Inverse-Scattering-Series is a multi-D series, the elimination algorithm/terms identified is a multi-D algorithm. Benefited from the previous work, we propose aInverse-Scattering-Series internal multiple elimination methodthat can eliminate all first-order internal multiples for all re-flectors for a multi–dimensional subsurface. Below shows a2D version of a higher order term in the elimination algorithm.

bE(ks,kg,qg +qs) =

+∞∫−∞

+∞∫−∞

dk1dk2

+∞∫−∞

dz1b1(kg,k1,z1)ei(qg+q1)z1

×z1−ε∫−∞

dz2F(k1,k2,z2)e−i(q1+q2)z2

+∞∫z2+ε

dz3b1(k2,ks,z3)ei(q2+qs)z3

(9)

F(k1,k2,z) =+∞∫−∞

d(q1 +q2)e−i(q1+q2)z+∞∫−∞

+∞∫−∞

dk′dk′′+∞∫−∞

dz′b1(k1,k′,z′)ei(q1+q′)z′

×z′+ε∫

z′−ε

dz′′g(k′,k′′,z′′)e−i(q′+q′′)z′′z′′+ε∫

z′′−ε

dz′′′g(k′′,k2,z′′′)ei(q′′+q2)z′′′

(10)

Internal Multiple Removal

g(k1,k2,z) =+∞∫−∞

d(q1 +q2)e−i(q1+q2)z+∞∫−∞

+∞∫−∞

dk′dk′′+∞∫−∞

dz′b1(k1,k′,z′)ei(q1+q′)z′

×z′+ε∫

z′−ε

dz′′b1(k′,k′′,z′′)e−i(q′+q′′)z′′z′′+ε∫

z′′−ε

dz′′′g(k′′,k2,z′′′)ei(q′′+q2)z′′′

(11)

Similar to the extension in the previous work in Zou and We-glein (2014) , F(k1,k2,z) and g(k1,k2,z) are two intermediatefunctions. Substituting into equation (9) provides a higher or-der term in the elimination sub-series.

Combining all of these kind of higher order terms provides theelimination algorithm in a 2D earth. The elimination algorithmfor a 3D earth is a straightforward extension.The complete sub-series is given in Zou et al. (2016).

CONCLUSION

The ISS internal multiple elimination algorithm is a part ofthe three-pronged strategy which is a direct response to cur-rent seismic processing and interpretation challenge when pri-maries and internal multiples are proximal to and/or interferewith each other in either on-shore and off-shore plays. This pa-per extends and generalizes the earlier 1D ISS internal multipleelimination method to provide an algorithm that eliminates allfirst-order internal multiples in a multi-D earth. This elimina-tion algorithm retains the stand-alone benefits of the ISS inter-nal multiple attenuation algorithm that can predict all internalmultiples at once (in contrast to stripping methods that removemultiples layer by layer and require subsurface information)and requiring no subsurface information.

ACKNOWLEDGMENTS

We are grateful to all M-OSRP sponsors for encouragementand support in this research. We would like to thank all ourcoworkers for their help in reviewing this paper and valuablediscussions.

Internal Multiple Removal

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