Adaptive Multiple Subtraction With a Pattern-Based Technique

7/27/2019 Adaptive Multiple Subtraction With a Pattern-Based Technique

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Adaptive Multiple Subtraction with a Pattern-based Technique Jiuying Guo*, Veritas DGC Inc

Summary

A pattern-based adaptive multiple subtraction technique is presented in this paper. It jointly uses the primary

prediction error filter(PEF) and its projection signal filter that is signal preserved as the constraints to calculate theadaptive matching filter in f-x domain. Both synthetic andreal data tests show its effectiveness.

Introduction

Surface-related multiple elimination(SRME) (Berkhout,

1982; Verschuur et al, 1992) has been an effectivealgorithm to remove multiples that are difficult to eliminateusing other conventional algorithms. SRME has two steps.The first step is multiple prediction, and the second step is

multiple adaptive subtraction. In the first step, the multiplemodel is constructed by 2D convolutions of the seismicdata without knowledge of the subsurface structures, and itcan handle complex structures. In reality, however,

variations in the acquisition wavelet, cable feathering, dipin crossline direction, boundary effect, limited offset rangecan introduce time shifts or amplitude artifacts into the

predicted multiples. Consequently, there are discrepancies between the real multiples and the predicted multiples. The

second step takes them into account. Conventional methodto do this is to apply a matching filter in t-x domain to the

predicted multiples, and then subtract the match-filtered

multiples from the seismic data. This matching filter iscalculated by a least squares algorithm. Unfortunately,when multiples strongly interfere with primaries, these

methods give biased primaries after subtraction.

Spitz(1999) presented a pattern-based algorithm, which is based on the popular assumption that the primaries and

multiples are predictable in f-x domain. It uses the prediction error filter(PEF) of the primaries as the

constraints of minimization to reduce the freedom of thesubtraction. This algorithm has proved to be particularlyefficient when attenuating the multiples in the mostcomplex structure areas where the multiples strongly

interfere with the primaries. Guitton et al(2001) introduceda similar algorithm in t-x domain. Abma et al (2002) madecomparisons of different adaptive subtraction techniques.However, in reality, the effectiveness of Spitz’s approach is

still somewhat limited when multiples are not perfectly predictable and strong random noises exist in the seismicdata.

In this paper, a more effective approach is presented. Itcalculates the matching filter by iteratively applying both

primary PEF and projection signal filter(Soubaras, 1994) as

the pattern constraints to the LSQR. Both synthetic and realdata tests show the results are encouraging.

Theory

• Surface-related multiple elimination

The first step of SRME is to predict the multiples based onthe feedback model. Its matrix notation is

( ) ( ) ......32

+−+−=∆ PPBPPBPBPPP (1)

B (2)11 −∧−= DR S

where is the input data, ∆ is the multiple-free data,

and is the seismic wavelet vector, is the receiver

matrix,

P P

S D∧R is the downward reflection matrix of the

surface.

The matrix multiplications are 2D convolution of the datawith itself in t-x domain. Due to various limitations, thereare discrepancies between the predicted multiples and the

real multiples as previously discussed. A matching filter has to be used to adaptively subtract the multiples.Unfortunately, when primaries strongly interfere withmultiples, conventional adaptive subtraction algorithms

give biased primaries. In order to solve the problem, aneffective algorithm is presented in this paper.

• Adaptive multiple subtraction using a pattern-based

technique

Linear events in f-x domain are predictable. A locally linear

event can be described by a local plane differentialequation

0=∂

∂+

∂

∂

t

P

x

P γ (3)

where ( )t x P , is the wavefield, and γ is local plane wave

slope. If γ is constant, the Fourier transform of (3) is

0ˆˆ

=+ P idx

P d γ ω (4)

The general solution of equation (4) is

( ) ( ) xie P x P

ωγ 0ˆˆ = (5)



Adaptive multiple subtraction with a pattern-based technique

where P ˆ is the Fourier transform of P . From (5), we can

easily see the predictability of the local linear event in f-x

domain. The prediction filter is , and its

corresponding PEF is

xie ωγ

( ) xie ωγ −,1 . For multi-events with

different slopes, the PEF is obtained by cascading severaltwo-term filters, and it can be represented by a polynomial.

The coefficients of the polynomial are the coefficients of

the data PEF . The Z transform of has the formd A d A

(6)(∏=

−=ld

j

jd d z z z A1

,)( )

( ) z Ad

( ) z A p

( ) z Ad

can be factored to the product of primary PEF

and multiple PEF . We can calculate

from the data, and from the predicted

multiples.

( ) z Am

( ) z Am

(7)( ) ( ) ( ) z A z A z A m pd =

where ( ) ( )∏=

−=

lp

j

j p p z z z A1

,

( ) ( )∏=

−=lm

j

jmm z z z A1

,

lmlpld +=

( ) z A p

can be obtained by deconvolving the ( ) z Ad with

.( ) z Am

The second step of SRME is adaptive multiple subtraction.Conventional way calculates the matching filter byminimizing the energy of the primaries. It can be

formulated as

Mf (8)0≈− P

(9)( ) PMMMf T T 1ˆ −

=

where is the matrix of multiples, f is the matrix of matching filters, f is the LSQR solution, and is thematrix of input data,

Mˆ P

T denotes the conjugate transpose.

When primaries strongly interfere with multiples, thisalgorithm gives biased primaries after subtraction. In order

to solve the problem linked with the standard LSQR

subtraction, Spitz proposed a method that uses the predicted multiple model as the input pattern. Then,

everything that resembles this pattern within a given subsetis subtracted from the data. This approach can be

formulated as

( ) 0≈− PMf A p(10)

( ) PAAMMAAM p

T

p

T

p

T

p

T 1ˆ −

=f (11)

where is the matrix of the primary PEFs. pA

The real data is always contaminated by additive randomnoises. In order to attenuate these noises, a signal-

preserving algorithm based on a projection filter wasintroduced(Soubaras,1994). In addition, multiples are notalways perfectly predictable in f-x domain. In order to

solve these problems, an algorithm that iteratively and jointly uses the primary PEF and the projection signal filter as the constraints in the minimization is proposed here.

Iteratively calculating the primary PEF can handle the problem when the multiples are not perfectly predictable,and using the projection signal filter can reduce the effectsof the additive random noises while preserving signals.This process can be formulated as

( ) 0≈− PMf BA p p(12)

( ) PBAABMMBAABMf p p

T

p

T

p

T

p p

T

p

T

p

T 1ˆ −

= (13)

where the matrix of projection signal filters is pB

( )22 −

+= IAAIB ε ε T

p p p

1,ε is a pre-whitening constant.

Examples

Both synthetic datasets (Courtesy of BP) and real datasets

have been successfully tested using this new algorithm. Fig.1 shows one flat primary event and two dipping multipleevents with different amplitudes. The amplitude of thesteepest event is twice of that of the other dipping event.

Fig. 2 shows the two dipping events that corresponding tothe multiples to be eliminated from the data. Fig. 3 showsthe result of the standard adaptive subtraction, which tellsus the multiples are only partially eliminated. Fig. 4 shows

the result of the new algorithm, which indicates that thetwo dipping events have been successfully removed. Fig. 5shows the events subtracted by the new algorithm. Fig. 6

shows the second synthetic data that consists of ahorizontal primary event, and a curved-multiple event. Fig.7 shows the predicted multiple that is slightly shifted with

respect to the curved event in Fig. 6. Fig. 8 is the result of




Spitz’s algorithm. It indicates that the primary is alsoattenuated due to that the curved event is not perfectly

predictable. Fig. 9 shows the events subtracted from thedata by Spitz’s algorithm. It’s obviously not what we

expected. Fig. 10 shows the result of the new algorithm.Clearly the multiples have been almost completelyremoved. Fig. 11 shows the events subtracted by the newalgorithm. Fig. 12 shows a common channel section with

multiples. Fig. 13 shows the predicted multiples. Fig. 14shows the result of standard adaptive multiple subtractionin t-x domain. We can see some residual multiples stillexist. Fig. 15 shows the result of the new algorithm. There

are no noticeable residual multiples. Fig. 16 shows thedifference between Fig.12 and Fig.15. It’s obvious that themultiples are almost perfectly removed.

Fig. 5: Events subtracted by the new algorithm

Fig. 6: Data

Fig. 1: Data

Fig. 7: Model(multiple)

Fig. 2: Model(multiples)

Fig. 8: Result of Spitz’s algorithm

Fig. 3: Result of standard adaptive subtraction

Fig. 9: Events subtracted by Spitz’s algorithm

Fig. 4: Result of the new algorithm




Fig. 10: Result of the new algorithm

Fig. 15: Result of the new adaptive subtraction algorithm

Fig. 11: Event subtracted by the new algorithm

Fig. 16: Difference between Fig. 12 and Fig. 15

Conclusions

A new algorithm of adaptive multiple subtraction with a pattern-based technique is introduced in this paper. Itcalculates the matching filter by jointly and iterativelyusing the signal PEF and the projection signal filter asconstraints to the LSQR problem. Both synthetic and real

data tests show its effectiveness. The extension of thealgorithm to 3D is straightforward.

Fig. 12: Common channel section

References

Abma, R., Kabir, N., Simon, A. S., McLain B., and MichellS., 2002, Comparisons of adaptive techniques for multiple

attenuation. SEG expanded abstracts.Berkhout, A. J., 1982, Seismic migration, imaging of acoustic energy by wave field extrapolation, A theoretical

aspects: Elsevier.Fig. 13: Predicted multiples

Guitton, A., Brown, M. Rickett, J., and Clapp, R., 2001,Multiple attenuation using t-x pattern-based subtractionmethod, 71st Ann. Internat. Mtg: Soc. of Expl. Geophys.,

1305-1308.Soubaras, R., 1994, Signal-preserving random noiseattenuation by the F-X projection: 64th Ann. Internat. Mtg,

Soc. Expl. Geophys., Expanded Abstract, 1576-1579.Spitz, S., 1999, Pattern recognition, spatial predictability,and subtraction of multiple events: The Leading Edge, 18,no.1, 55-58.Verschuur, D. J., Berkhout, A. J., and Wapenaar, C. P. A.,

1992, Adaptive surface-related multiple attenuation:Geophysics, 57, 1166-1177.

Fig. 14: Standard adaptive multiple subtraction in t-x

domain

Adaptive Multiple Subtraction With a Pattern-Based Technique

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