An analytic example examining the reference velocity sensitivity of the elastic internal multiple attenuation algorithm S.-Y. Hsu, S. Jiang and A. B. Weglein, M-OSRP, University of

An analytic example examining the reference velocity sensitivityof the elastic internal multiple attenuation algorithm

S.-Y. Hsu, S. Jiang and A. B. Weglein, M-OSRP, University of Houston

Abstract

Internal multiple attenuation is a pre-processing step for seismic imaging and amplitude analysis.Hsu and Weglein (2007) showed that the internal multiple algorithm from the inverse scatteringseries is independent of reference velocity for an 1D earth with acoustic background. In thisstudy, we followed the analysis of Nita and Weglein (2005) and used the elastic internal multiplealgorithm (Matson, 1997) to investigate velocity sensitivity for an 1D earth with elastic referencemedium. The result suggests that the algorithm is also independent of reference velocity for an1D earth with elastic background.

1 Introduction

Seismic processing is used to estimate subsurface properties from the reflected wave-fields. Theseismic data are a set of reflected waves including primaries and multiples. Primaries are eventsthat have only one upward reflection before arriving at the receiver. Multiply reflected events(multiples) are classified as free-surface or internal multiples depending on the location of theirdownward reflections. Free-surface multiples have at least one downward reflection at the air-water or air-land surface (free surface). Internal multiples have all downward reflections below themeasurement surface (Weglein et al., 1997; Weglein and Matson, 1998).

Methods for seismic imaging and amplitude analysis usually assume that the reflection data areprimaries-only. To accommodate this assumption, multiple removal/attenuation is a prerequisitefor seismic processing. Conventional methods successfully attenuate multiples by assuming simpleor known subsurface geology. However, in geologically complex areas those methods may becomeinadequate (Otnes et al., 2004). To overcome the limitations of conventional methods, the inversescattering series (ISS) methods were proposed to perform multiple removal/suppression for acousticdata without subsurface information or assumptions (Carvalho, 1992; Araujo, 1994; Weglein et al.,1997). Following this framework, Matson (1997) extended the multiple attenuation algorithm fromacoustic to elastic.

In order to keep the perturbation below the measurement surface, the ISS multiple removal/attenuationmethods usually require a known source wavelet and information about the near surface (Matson,1997). To obtain the source wavelet, Wang and Weglein (2008) has shown how Greens theorem isused to fulfill the requirement. The need of the near surface properties is a practical obstacle for on-shore/ocean bottom application. Here, we present an analytic example to show the inner workingsof the elastic internal multiple algorithm. In particular, we demonstrate how this scattering-basedalgorithm can predict exact arrival times without the above assumption being true.

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Internal multiple reference velocity sensitivity MOSRP08

2 Theory

We start with the forward scattering series derived from the Lippmann-Schwinger equation

G = G0 +G0V G, (1)

which can be expanded in a forward series

G = G0 +G0V G0 +G0V G0V G0 + , (2)

where G and G0 are the actual and reference Greens functions. The perturbation V can be writtenas V =

n Vn where Vn is n-th order in the data.

Define D = GG0 as the measurement of the scattered field. Substituting into the forward seriesgives the inverse scattering series in terms of data

D = G0V1G0,0 = G0V2G0 +G0V1G0V1G0,0 = G0V3G0 +G0V2G0V1G0 +G0V1G0V2G0 +G0V1G0V1G0V1G0,....

(3)

The term G0V1G0V1G0V1G0 is the first term in the inverse series of first order internal multiples.

Following the method developed by Weglein et al. (1997) and Araujo (1994), the equation for firstorder internal multiples in 2-D acoustic data is

b3IM (kg, ks, qg + qs) =1

(2pi)2

dk1eiq1(zgzs)

dk2eiq2(zgzs)

dz1b1(kg, k1, z1)e

i(qg+q1)z1

z1

dz2b1(k1, k2, z2)e

i(q1+q2)z2

z2+

dz3b1(k2, ks, z3)e

i(q2+qs)z3

(4)

The parameter > 0 ensures z1 > z2 and z3 > z2 which satisfy the geometric relationship betweenreflections of internal multiples (lower-higher-lower). b1 is defined in terms of the original pre-stackdata without free surface multiples. The data can be written as

D(kg, ks, ) = (2iqs)1b1(kg, ks, qg + qs), (5)

where b1(kg, ks, qg + qs) represents the data result from a single-frequency incident plane wave.

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The adapted elastic version of (4) given by Matson (1997) is

B3IMij (kg, ks, qig + q

js) =

1(2pi)2

dk3eiql1(zgzs)

dk2eiqm2 (zgzs)

dz1Bil(kg, k1, z1)e

i(qig+ql1)z

1

z1

dz2Blm(k1, k2, z2)e

i(ql1+qm2 )z2

z2+

dz3Bmj(k2, ks, z3)e

i(qm2 +qjs)z

3

(6)

where qi1 indicate P and S vertical wavenumbers for i = P and S, respectively.

Similarly, Bij is defined in terms of the original pre-stack data, that is

Dij(k1, k2, ) = (2iqj2)1Bij(kg, ks, qi1 + qj2). (7)

Hence,

DPP (k1, k2, ) =(2iqP2 )1BPP (kg, ks, qP1 + qP2 ),DPS(k1, k2, ) =(2iqS2 )1BPS(kg, ks, qP1 + qS2 ),DSP (k1, k2, ) =(2iqP2 )1BSP (kg, ks, qS1 + qP2 ),DSS(k1, k2, ) =(2iqS2 )1BSS(kg, ks, qS1 + qS2 ).

(8)

One can see that if the converted waves do not exist, equation (6) becomes equation (4), which isthe first order internal multiple attenuator in acoustic form.

3 Attenuation of elastic internal multiples : a 1.5D example

We consider a layering model given by Nita and Weglein (2005) (see Figure 1). The velocity changesacross the interfaces located at z = za and z = zb where the velocities are c0, c1, and c2, respectively.The sources and receivers are located at the measurement surface where the depth z = 0. Thereference vertical speeds are defined as

civ ={0/cosi, if i = P0/cosi, if i = S

(9)

The reference horizontal speed is defined as

cih ={0/sini, if i = P0/sini, if i = S

(10)

where 0 and 0 are P -wave and S-wave velocity in the reference medium, respectively.

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Figure 1: The model for the 1.5D example

Figure 2: The geometry of the first primary in the data

The horizontal wave numbers are

kis = /cih,

kjg = /cjh.

(11)

For media with depth-dependent velocity, the horizontal slowness (sini/c(z) p) is constant along

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a ray path whether or not converted waves are present (Aki and Richards, 2002). Therefore,

kis = kjg = /ch,

kis + kjg = 2/ch.

(12)

The vertical wave numbers are

qis =

( 0 )

2 k2s , if i = P( 0 )

2 k2s , ifi = S(13)

qjg =

( 0 )

2 k2g , if j = P( 0 )

2 k2g , ifj = S(14)

The total travel time for the n-th primary can be written as

Tn = n + th, (15)

where n and th are the vertical and horizontal travel times, respectively. The horizontal traveltime is defined as

th =xg xsch

=2xhch

, (16)

and the vertical travel time corresponding to the n-th event is

n =znciv

+zn

cjv, (17)

where zn is the pseudo depths for the n-th event. Therefore,

n = zn(

civ+

cjv)

= zn(qis + qjg) = znk

ijz ,

th = (2xhch

) = khxh.

(18)

The data in the frequency domain can be written as

Dij(xh, 0;) =12pi

dkhn

Rijn eikijz zn

2iqiseikhxh . (19)

where kijz = qis + qjg, kh = kg + ks ,and xh = (xg xs)/2.

Fourier transform over xg, xs,

Dij(kh, 0, ) =2pi2iqis

n

Rijn eiznk

ijz (kg ks). (20)

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The data is

Bij(kh, 0, ) = 2iqisDij(kh, 0, )

= 2pin

Rijn eikijz zn(kg ks). (21)

Inverse Fourier transforming into the pseudo depth domain gives

Bij(kh, z, ) = 2pin

Rijn (z zn)(kg ks). (22)

Substituting equation (22) into equation (6), the integration over z3 gives z2+

dz3eikmjz z3Bmj(k2, k

s, z

3) =2pi

z2+

dz3eikmjz z3

p

Rmjp (z3 zp)(k2 ks)

=2pip

Rmjp eikmjz zpH(zp (z2 + ))(k2 ks)

The integration over z2 is z1

dz2eiklmz z2Blm(k1, k

2, z

2) 2pi

p


= (2pi)2 z1

dz2eiklmz z2

q

Rlmq (z2 zq)(k1 k2)

p


= (2pi)2p,q

Rmjp Rlmq e

ikmjz zpeiklmz z

qH(zp (zq + ))H((z1 ) zq)(k2 ks)(k1 k2)

The integration over z1 is

dz1eikjlz z1Bjl(kg, k

1, z

1)

[(2pi)2p,q

Rmjp Rlmq e

ikmjz zpeiklmz z

qH(zp (zq + ))H((z1 ) zq)(k2 ks)(k1 k2)]

=(2pi)3

dz1eikjlz z1

r

Rjlr (z1 zr)(kg k1)

p,q

Rmjp Rlmq e

ikmjz zpeiklmz z

q

H(zp (zq + ))H((z1 ) zq)(k2 ks)(k1 k2)=(2pi)3

p,q,r

Rmjp Rlmq R

jlr e

i(kmjz zpklmz zq+kjlz zr)H(zp (zq + ))H(zr (zq + ))

(k2 ks)(k1 k2)(kg k1)

(23)

Note that the parameter > 0 ensures zp > zq and zr > zq which satisfy a lower-higher-lowerrelationship between the pseudo depths.

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The result for B3IMij is

B3IMij (kh, z, ) = 2pip,q,rzp>zqzr>zq

Rmjp Rlmq R

jlr e

i(kmjz zpklmz zq+kjlz zr)(kg ks) (24)

Inverse Fourier transforming over km and kh gives

B3IMij (kh, k

z, ) =

12pi

dkh

p,q,r

Rmjp Rlmq R

jlr e

i(kmjz zpklmz zq+kjlz zr)eikhxh . (25)

The predicted first internal multiple in the space domain is

IM1stpredicted =12pi

dkh

p,q,r

Rmjp Rlmq Rjlr e

i(kmjz zpklmz zq+kjlz zr)eikhxh

2iqis. (26)

Note that

M.S.

Figure 3: The geometry of pseudo depths

kmjz zp = p,klmz zq = q,kjlz zr = r,khxh = th.

(27)

Therefore,

kmjz zp klmz zq + kjlz zr + khxh = ( p q + r + th) (28)

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The travel time for the predicted first order internal multiple is

p q + r + th = (Tp th) (Tq th) + (Tr th) + th= Tp Tq + Tr (29)

where Tp, Tq and Tr are travel times corresponding to the p-th, q-th, and r-th event, respectively.The travel time for the actual first order internal multiple is

Ttotal = p q + r + th = (Tp th) (Tq th) + (Tr th) + th= Tp Tq + Tr. (30)

The result Ttotal = Tp Tq + Tr agrees with the actual arrival time for the first order internalmultiple.

One can see that reference velocity errors give incorrect vertical and horizontal travel times for eachevent (see equation (16) and (17)). However, equation (29) shows that the errors due to wrongreference velocities can be canceled, and the predicted arrival times remain correct.

To attenuate an internal multiple for a single frequency requires data at all frequencies. Thisrequirement comes from the integral over temporal frequency when transforming data from thefrequency domain to the pseudo-depth domain. The integral transformation is truncated with band-limited data. The 1.5D example suggests that the algorithm is insensitive to reference velocitiesand hence has potential for spectral extrapolation if we choose reasonable reference velocities.

4 Conclusion

In this paper, we have presented an elastic internal multiple attenuation algorithm (Matson, 1997)following the analysis of Nita and Weglein (2005). Although the scattering-based methods usuallyrequire known near surface properties, our study has suggested that the internal multiple attenu-ation algorithm can tolerate errors in reference velocity for 1D earth. The capability of predictingexact travel time without known reference velocities is useful for land application, where the nearsurface properties are often complicated and ill-defined. Moreover, the freedom of choosing ref-erence velocities has potential for the spectral extrapolation. The future direction of this studyincludes extending the algorithm to multi-dimensional elastic case and extrapolation of band-limitedsynthetic data.

Acknowledgments

We would like to thank all M-OSRP sponsors for constant support and encouragement. We espe-cially thank all the members in M-OSRP for valuable suggestions.

References

Aki, K. and P. G. Richards. Quantitative Seismology. 2nd edition. University Science Books, 2002.

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Araujo, F. V. Linear and non-linear methods derived from scattering theory: backscattered tomog-raphy and internal multiple attenuation. PhD thesis, Universidade Federal da Bahia, 1994.

Carvalho, P. M. Free-surface multiple reflection elimination method based on nonlinear inversionof seismic data. PhD thesis, Universidade Federal da Bahia, 1992.

Hsu, Shih-Ying and Arthur B. Weglein. On-shore project report I Reference velocity sensitivityfor the marine internal multiple attenuation algorithm: analytic examples. Technical report,Mission-Oriented Seismic Research Project, University of Houst on, 2007.

Matson, K. H. An inverse-scattering series method for attenuating elastic multiples from multi-component land and ocean bottom seismic data. PhD thesis, University of British Columbia,1997.

Nita, Bogdan G. and Arthur B. Weglein. Inverse scattering internal multiple attenuation algo-rithm in complex multi-d media. Technical report, Mission-Oriented Seismic Research Project,University of Houston, 2005.

Otnes, Einar, Ketil Hokstad, and Roger Sollie. Attenuation of internal multiples formulticomponent- and towed streamer data.. SEG Technical Program Expanded Abstracts 23(2004): 12971300.

Wang, Min and Arthur B. Weglein. A short note about elastic wavelet estimation. Technical report,Mission-Oriented Seismic Research Project, University of Houst on, 2008.

Weglein, A. B., F. A. Gasparotto, P. M. Carvalho, and R. H. Stolt. An Inverse-Scattering SeriesMethod for Attenuating Multiples in Seismic Reflection Data. Geophysics 62 (November-December 1997): 19751989.

Weglein, Arthur B. and Ken H. Matson. Inverse scattering internal multiple attenuation: ananalytic example and subevent interpretation. SPIE, 1998, 108117.

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An analytic example examining the reference velocity sensitivity of the elastic internal multiple attenuation algorithm S.-Y. Hsu, S. Jiang and A. B. Weglein, M-OSRP, University of