Zhang-Weglein-2005.pdf

The inverse scattering series for tasks associated with primaries: Depth imaging anddirect non-linear inversion of 1D variable velocity and density acoustic mediaHaiyan Zhangand Arthur B. Weglein, University of Houston

Summary

This paper presents the rst analysis, and direct depthimaging and inversion algorithms for tasks associatedwith primaries for a vertically varying velocity anddensity acoustic medium. The method is derived fromthe inverse scattering series, and, hence, assumes theactual subsurface properties governing the propagationof waves is neither known nor determined. The terms inthe series that correspond to tasks for imaging-only andinversion-only are identied and separated. Tests withanalytic data indicate signicant added value, beyondlinear estimates, in terms of both the proximity to actualproperties and the increased range of angles over whichthe improved estimates are valid.

Introduction

The ultimate objective of inverse problems is to deter-mine medium and target properties from reectiondata. However, that objective has never been achievedin a straight ahead single step manner. The onlydirect multi-dimensional inversion procedure for seismicapplication, is the task specic subseries of the inversescattering series. (Weglein et al., 2003). The order ofprocessing tasks is (1) free surface multiple removal;(2) internal multiple removal; (3) imaging primariesin space and (4) inversion. The free surface multipleremoval and internal multiple attenuation subseries havebeen presented by Weglein et al. (1997). Those twomultiple removal procedures are model type independent.Taking internal multiple algorithm from attenuationto elimination is being studied (Ramirez and Weglein,2005). In contrast with model type independent multipleremoval procedures, there is a full expectation that tasksand algorithms associated with primaries will have acloser interest in model type. For example, there is noway to even imagine that medium property identicationcan take place without reference to a specic model type.A staged approach and isolation of tasks philosophy isessential in this yet tougher neighborhood of primaries.The stages within the strategy for primaries are asfollows: (1) 1D earth, with one parameter, velocity as afunction of depth, and a normal incidence wave (Shaw etal., 2003); (2) 1D earth with one parameter subsurfaceand oset data, one shot record (Shaw, 2005); (3) 2Dearth with one parameter, velocity, varying in x and z,and a suite of shot records (Liu et al., 2005); (4) 1Dacoustic earth with two parameters varying, velocityand density, one propagation velocity, and one shotrecord of PP data, and (5) 1D elastic earth, two elastic

1617 Science & Research Bldg 1, Houston, TX 77204-5005

isotropic parameters and density, and two wave speeds,for P and S waves, and PP, PS, SP, and SS shot recordscollected (Zhang et al., 2005). This paper deals withstage (4). The model is acoustic and a second paper inthis progression and evolution generalizes this for theelastic case.

In this paper, the rst direct non-linear inversion methodis obtained for 1D two-parameter acoustic media and a2D experiment. From this solution, the tasks for imaging-only and inversion-only terms are successfully separated.Tests with analytic data indicate signicant added value,beyond linear estimates, in terms of both the proximity toactual value and the increased range of angles over whichthe improved estimates are provided. The depth imagingalgorithm without the velocity rst decides if the inputvelocity is adequate, and if adequate conventional migra-tion is prescribed. If the data decides the verdict on thevelocity is inadequate it acts to remove the incorrect im-age and constructs the correct one, without knowing ordetermining the velocity. Other terms in the algorithmare identied as performing non-linear direct AVO, andonce again, allow the data self determination of overbur-den velocity adequacy, and acts accordingly to improveupon linear estimates of property changes. The role of ve-locity is claried, as central and all-important in location,in that only an incorrect velocity causes a depth imagingresponse from the series, independent of how you param-eterize the acoustic problem, or what error estimates ofother properties might suer. Benet of non-linear directinversion is demonstrated over linear standard procedure,for a set of examples using analytic and numerical tech-niques. The common problem of linear leaking betweenlinear property change predictions is also addressed by theseries, and located and analyzed in this paper.

The paper has the following structure: In section 2 weshow the derivation in detail and then discuss a specialparameter followed by numerical tests. The last sectionhas conclusions and acknowledgements.

Derivation of 1, 1 and 2, 2

To illustrate task (4), target identication, we considera 1D acoustic two-parameter earth model (e.g. bulkmodulus and density or velocity and density). We startwith the 3D acoustic wave equations in the actual andreference medium (Clayton and Stolt, 1981 and Wegleinet al., 1997)

[2

K(r)+ 1

(r)]G(r, rs;) = (r rs), (1)

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Direct non-linear two parameter 2D acoustic inversion[2

K0(r)+ 1

0(r)]G0(r, rs;) = (r rs), (2)

where G(r, rs;) and G0(r, rs;) are respectively thecausal Greens operators that describe wave propagationin the actual and reference medium. The bulk modulus,K, is given by c2, where c is P-wave velocity and is thedensity. Then the perturbation can be written as (We-glein et al., 2003)

V =2

K0+

0, (3)

where = 1 K0K

and = 1 0

are the two parameters

we choose to perform the following non-linear inversion.Assuming both 0 and c0 are constants, and for a 1-Dacoustic medium, we expand V (z,), (z) and (z) inthe same way as (Weglein et al., 1997)

V = V1 + V2 + , (4)

we have

V1(z,) = 21(z)

K0+

1

01(z)

2

x2+

1

0

z1(z)

z, (5)

V2(z,) = 22(z)

K0+

1

02(z)

2

x2+

1

0

z2(z)

z, (6)

...,

where the subscript i in Vi, i and i denote the portionof those quantities i-th order in the data. Substituting(5) into the rst equation of the inverse scattering series(Weglein et al., 2003) D = [G0V1G0]ms, we can obtainthe linear solution for 1 and 1 in the frequency domain

D(qg, , zg, zs) = 04

eiqg(zs+zg)

[

1

cos2 1(2qg) + (1 tan2 )1(2qg)

], (7)

where the subscripts s and g denote source and receiverquantities respectively, and qg, and k = /c0 shownin Figure 1, have the following relations (Matson, 1997):qg = qs = k cos , kg = ks = k sin . Similarly, substitut-

gq k

gk

z

000,, Kc

Kc ,,

Fig. 1: The relationship between qg , kg and .

ing (6) into the second equation of the inverse scattering

series [G0V2G0]ms = [G0V1G0V1G0]ms (Weglein et al.,2003), we derive the solution for 2(z) and 2(z) as afunction of 1(z) and 1(z)

1

cos2 2(z) + (1 tan2 )2(z)

= 12 cos4

21(z) 12(1 + tan4 )21(z) +

tan2

cos2 1(z)1(z)

12 cos4

1(z)

z0

dz[1(z) 1(z)]

+1

2(tan4 1)1(z)

z0

dz[1(z) 1(z)], (8)

where 1(z) =d1(z)

dz, 1(z) =

d1(z)dz

.

We have obtained in equation (8) the rst two parameterdirect non-linear inversion of 1D acoustic media for a 2Dexperiment. Equations (7) and (8) imply that two dier-ent angles , can determine 1, 1 and then 2, 2. Fora single-interface example, we can also show that onlythe rst three terms on the right hand side contributeto amplitude correction, while the last two terms per-form imaging in depth. Therefore, in this way, the tasksfor imaging-only and inversion-only terms are identiedand examined. If another choice of free parameter otherthan (e.g., or kh) was selected, then the functionalform between the data and the rst order perturbationchanges. Furthermore, the relationship between the rstand second order perturbation is then also dierent, andnew analysis would be required for the purpose of iden-tifying specic task separated terms. In our experience,the choice of as free parameter (for a 1D medium) isparticularly well suited for allowing a task separated iden-tication of terms in the inverse series. Details about thesignicance of this solution will be presented in the fol-lowing sections.

Special parameter for linear inversion

As we mentioned above, since the relationship betweendata and target property changes is non-linear, linearinversion will produce errors in target property pre-diction. When an actual property change is zero, thelinear prediction of the change can be non-zero. Also,when the actual change is positive, the predicted linearapproximation can be negative. There is a specialparameter for linear inversion of acoustic media, thatnever suers the latter problem the P-wave velocity.

From the one interface case, we can show that when c0 =c1, i.e., when c = 0, (c)1 = 0. This generalizes towhen c > 0, then (c)1 > 0, or when c < 0, then(c)1 < 0, as well.

Therefore, we can rst obtain the right sign of relativechange in P-wave velocity from the linear inversion (c)1,then, we can obtain a more accurate amplitude when weinclude non-linear terms. The linear approximation tothe change in velocity,

(cc

)1, is 1

2(1 1).

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Direct non-linear two parameter 2D acoustic inversion

We also note that when the velocity doesnt change acrossan interface, i.e., c0 = c1, looking at the integrand ofimaging terms 1 1 (see (8)), the image doesnt movebecause 1 1 = 0 in this situation. We can see thismore explicitly when we change the two parameters and to c

cand , and we can rewrite (8) as

1

cos2

(c

c

)2

(z) + 2(z)

=cos2 22 cos4

(c

c

)21

(z) 1221(z)

1cos4

(c

c

)1

(z)

z0

dz(

c

c

)1

1cos2

1(z)

z0

dz(

c

c

)1

. (9)

This equation makes two key points. One is that there isno leakage correction at all in this expression. (Leakage isa two parameter eect and its removal is associated witha cross term, like the 11 term in equation (8)). Theabsence of a cross term in equation (9) is an indicationof the special leakage resistant properties of the P-wavevelocity.

The second point is that, when we look at the integrand(cc

)1of the imaging terms, it indicates that if we have

the right velocity, the imaging terms will automaticallybe zero and no integration is performed. On the otherhand, if we didnt have the right velocity, these imagingterms would move the interface from the wrong locationcloser to the right location. The conclusion is that thedepth imaging terms depend only on the velocity errors.

Numerical tests

Consider the one interface example (in Figure 2), and

000,, Kc

111,, Kc

z

x

a

0

Fig. 2: 1D one interface acoustic model.

assume the interface surface is at z = a, and supposezs = zg = 0. The reection coecient has the followingform (Keys, 1989)

R() =(1/0)(c1/c0)

1 sin2

1 (c21/c20) sin2

(1/0)(c1/c0)

1 sin2 +

1 (c21/c20) sin2 ,

(10)where , the angle of P-wave incidence, is same as thatin equation (7) and (8), and 0 and 1 denote the density

for the reference medium and actual medium respectively.Then using perfect data (Clayton and Stolt, 1981 andWeglein et al., 1986)

D(qg, ) = 0R()e2iqga

4iqg, (11)

and substituting (11) into (7), after Fourier transforma-tion over 2qg, for z > a and xed , we obtain

1

cos2 1(z) + (1 tan2 )1(z) = 4R()H(z a). (12)

In this case of a single reector, the non-linear equation(8) reduce to

1

cos2 2(z) + (1 tan2 )2(z)

= 12 cos4

21(z) 12(1 + tan4 )21(z)

+tan2

cos2 1(z)1(z), (13)

From equations (12) and (13) we choose two dierent an-gles to solve for 1 and 1, and then 2 and 2.

For a specic model, 0 = 1.0g/cm3, 1 = 1.1g/cm

3, c0 =1500m/s and c1 = 1700m/s. In the following gures, wepresent the results for the relative changes in the P-wave

bulk modulus( = K

K

), density

( =

), impedance(

II

)and velocity

(cc

)corresponding to dierent pairs

of 1 and 2.

From Figure 3, we notice that when we add 2 to 1, theresult is much closer to the exact value of . Furthermore,the result is better behaved, i.e., the plot surface becomesatter, over a larger range of precritical angles. Similarly,from Figure 4, we can also see the results of 1 + 2 aremuch better than those of 1. In addition, we observe thatthe sign of the linear approximation to the relative changein density, 1, is wrong at some angles, while, the resultsfor 1 + 2 always have the right sign. After including2, we correct the sign of the density change, which isvery important in the generalization to identication ofan elastic earth.

Conclusion

Including terms beyond linear in the earth property iden-tication subseries provides added value. Although themodel we used in the numerical test is simple, equations(7) and (8) are also generalizable for multidimensionalmedia and complex targets. The inverse scatteringseries is a direct inversion procedure which inverts dataindependent of the properties of the target, withoutassumptions such as smooth geometry or small contrast.This work is a major step towards the realism for targetidentication. The numerical results are encouragingand this work will be extended to study the elastic caseusing three parameters (see, e.g., Boyse, 1986 and Boyseand Keller, 1986).

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Direct non-linear two parameter 2D acoustic inversion

Acknowledgements

We would like to thank R. Keys, D. Foster, R. Stolt, B.Nita, S. Shaw and K. Innanen for valuable discussionsand useful suggestions. The support of the sponsorsof M-OSRP and the support from NSF-CMG awardnumber DMS-0327778 are gratefully acknowledged.

References

Boyse W E 1986 Wave propagation and inversion inslightly inhomogeneous media p 40

Boyse W E and Keller J B 1986 Inverse elastic scatteringin three dimensions J. Acoust. Soc. Am. 79 215218

Clayton R W and Stolt R H 1981 A Born-WKBJ inversionmethod for acoustic reection data for attenuatingmultiples in seismic reection data Geophysics 4615591567

Innanen K A 2003 Methods for the Treatment of Acous-tic and Absorptive/Dispersive Wave Field Measure-ments Ph.D. thesis University of British Columbia

Keys R G 1989 Polarity reversals in reections from lay-ered media Geophysics 54 900905

Liu F, Weglein A B, Innanen K A and Nita B G 2005Inverse scattering series for vertically and laterallyvarying modia: application to velocity independentdepth imaging M-OSRP Annual Report 2004 176-263preparing for publication

Matson K H 1997 An inverse scattering series methodfor attenuating elastic multiples from multicompo-nent land and ocean bottom seismic data Ph.D. the-sis University of British Columbia p 18

Ramirez A C and Weglein A B 2005 An inverse scatteringinternal multiple elimination method: Beyond atten-uation, a new algorithm and initial tests M-OSRPAnnual Report 2004 138-157 preparing for publica-tion

Shaw S A, Weglein A B, Foster D J, Matson K H and KeysR G 2003 Isolation of a leading order depth imag-ing series and analysis of its convergence properties,M-OSRP Annual Report 2002 157-195 preparing forpublication

Shaw S A 2005 An inverse scattering series algorithmfor depth imaging of reection data from a layeredacoustic medium with an unknown velocity modelPh.D. thesis University of Houston

Weglein A B, Violette P B and Keho T H 1986 Usingmultiparameter Born theory to obtain certain exactmultiparameter inversion goals Geophysics 51 10691074

Weglein A B, Gasparotto F A, Carvalho P M and Stolt RH 1997 An inverse-scattering series method for atten-uating multiples in seismic reection data Geophysics62 19751989

Weglein A B, Araujo F V, Carvalho P M, Stolt R H,Matson K H, Coates R, Corrigan D, Foster D J, ShawS A and Zhang H 2003 Inverse scattering series andseismic exploration Inverse Problem 19 R27R83

Zhang H, Weglein A B and Keys R G 2005 Velocity in-dependent depth imaging and non-linear direct tar-get identication for 1D elastic media: testing andevaluation for application to non-linear AVO, usingonly PP data M-OSRP Annual Report 2004 312-338preparing for publication

010

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4050

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theta2

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1+

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theta2

theta1

Fig. 3: 1 (left) and 1 + 2 (right) displayed as a functionof two dierent angles. The red line in the gures present theexact value of . In this example, the exact value of is 0.292.

010

2030

40

50

-0.02-0.010.000.010.020.030.040.050.060.070.080.090.100.110.120.130.140.15

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theta2

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Fig. 4: 1 (left) and 1+2 (right). In this example, the exactvalue of is 0.09.

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2005 SEG Technical Program Expanded Abstracts have been copy edited 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. The inverse scattering series for tasks associated with primaries: Depth imaging and direct non-linear inversion of 1D variable velocity and density acoustic media References Boyse, W. E., 1986, Wave propagation and inversion in slightly inhomogeneous media:

40. Boyse, W. E., and J. B. Keller, 1986, Inverse elastic scattering in three dimensions:

Journal of the Acoustical Society of America, 79 215218. Clayton, R. W. and R. H. Stolt, 1981, A born-WKBJ inversion method for acoustic

reflection data for attenuating multiples in seismic reflection data: Geophysics, 46 15591567.

Innanen, K. A., 2003, Methods for the treatment of acoustic and absorptive/dispersive wave field measurements: Ph.D. thesis, University of British Columbia.

Keys, R. G., 1989, Polarity reversals in reflections from layered media: Geophysics, 54 900905.

Liu, F., A. B. Weglein, K. A. Innanen, and B. G. Nita, 2005, Inverse scattering series for vertically and laterally varying media: Application to velocity independent depth imaging: Mission-Oriented Seismic Research Program Annual Report, 176263.

Matson, K. H., 1997, An inverse scattering series method for attenuating elastic multiples from multicomponent land and ocean bottom seismic data: Ph.D. thesis, University of British Columbia.

Ramirez, A. C., and A. B. Weglein, 2005, An inverse scattering internal multiple elimination method: Beyond attenuation, a new algorithm and initial tests: Mission-Oriented Seismic Research Program Annual Report, 138157.

Shaw, S. A., 2005, An inverse scattering series algorithm for depth imaging of reflection data from a layered acoustic medium with an unknown velocity model: Ph.D. thesis, University of Houston.

Shaw, S. A., A. B. Weglein, D. J. Foster, K. H. Matson, and R. G. Keys, 2003, Isolation of a leading order depth imaging series and analysis of its convergence properties: Mission-Oriented Seismic Research Program Annual Report, 157195.

Weglein, A. B., F. A. Arajo Gasparotto, P. M. Carvalho, and R. H. Stolt, 1997, An inverse-scattering series method for attenuating multiples in seismic reflection data: Geophysics 62, 19751989.

Weglein, A. B., F. V. Arajo Gasparotto, P. M. Carvalho, R. H. Stolt, K. H. Matson, R. Coates, D. Corrigan, D. J. Foster, S. A. Shaw, and H. Zhang, 2003, Inverse scattering series and seismic exploration: Inverse Problems, 19, R27R83.

Weglein, A. B., P. B. Violette, and T. H. Keho, 1986, Using multiparameter Born theory to obtain certain exact multiparameter inversion goals: Geophysics, 51 10691074

Zhang, H., A. B. Weglein, and R. G. Keys, 2005, Velocity independent depth imaging and non-linear direct target identification for 1D elastic media: testing and

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evaluation for application to non-linear AVO, using only PP data: Mission-Oriented Seismic Research Program Annual Report, 312338.

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