The inverse scattering series for tasks associated with primaries: direct non-linearinversion of 1D elastic media

Haiyan Zhang∗and Arthur B. Weglein,M-OSRP, Dept. of Physics, University of Houston

Summary

In this paper, research on direct inversion for two pa-rameter acoustic media (Zhang and Weglein, 2005) isextended to the three parameter elastic case. We presentthe first set of direct non-linear inversion equations for1D elastic media (i.e., depth varying P-velocity, shearvelocity and density). The terms for moving mislocatedreflectors are shown to be separable from amplitudecorrection terms. Although in principle this directinversion approach requires all four components of elasticdata, synthetic tests indicate that consistent value-added

results may be achieved given only DPP measurements.We can reasonably infer that further value would derive

from actually measuring DPP , DPS , DSP and DSS asthe method requires. The method is direct with neithera model matching nor cost function minimization.

Introduction

The objective of seismic exploration is to predict thelocation and properties of the hydrocarbon resources inthe earth (i.e., imaging and inversion) using recordedseismic data. The character (i.e., the amplitude andphase) of the reflected data depends on the propertiesof the medium that the wave travels through, and thecontrasts in properties that cause those reflections. Thereflection process has a non-linear dependence on anyproperty change at the reflector. Current inversion meth-ods either assume a simple linear relationship and solvean approximate form, or assume a non-linear relationshipbut invoke an indirect method (e.g., minimization of anobjective function) to do the inversion. The assumptionsof the former methods are often violated in practiceand can cause erroneous predictions; the latter cate-gory usually involve big computation effort and/or hasambiguity issues in the predicted result. In this paper,a new method based on direct non-linear inversion isdeveloped and analyzed. The procedure is derived asa task-specific subseries (see, e.g., Weglein et al., 2003)of the inverse scattering series (ISS). To date, this isthe only candidate method with more realistic, morephysically complete and hence, more reliable predictioncapability and potential. What makes the task specificsubseries methods powerful is that each subseries has lessto achieve and hence better convergence properties thanthe full series. The original ISS research, aimed at freesurface multiple removal and internal multiple removal,resulted in successful application on field data (Wegleinet al., 1997 and Weglein et al., 2003). The next step is

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

the processing of primaries.

Beginning in 2001, tasks that work on primaries, whichinclude depth imaging and parameter estimation, wereinvestigated and analyzed. Single parameter 1D acousticconstant density media and 1D normal incidence frame-works were broached first (Weglein et al., 2003; Innanen,2003; Shaw, 2005). Extension then took two forms: toa one parameter 2D acoustic medium (Liu et al., 2005),and to a two parameter 1D acoustic medium (Zhang andWeglein, 2005). In this paper we develop the inversionequations for a three parameter 1D elastic medium, anon-trivial step towards realism in the exploration seismicframework. We take these steps in 1D to allow the use ofanalytic data for numerical tests, and to prime the nextstep: extension to a multi-parameter, multi-dimensionalmedium.

The non-linear direct elastic inversion method describedin this paper requires four data types, DPP , DPS , DSP

and DSS as input. A major theme here is to show

how DPP can be used to approximately synthesize the

DPS, DSP and DSS such that high quality inversion re-sults can still be achieved with the measurement of onlyone data type. This permits us to perform elastic in-version using only pressure measurements, i.e., towedstreamer data.

In the following we first briefly review elastic inverse scat-tering theory and then present the solutions and numeri-

cal tests for non-linear inversion when only DPP is avail-able.

Background

In this section we consider the inversion problem in twodimensions for an elastic medium. For convenience, wechange the basis and transform the equations of displace-ment space into PS space. In the PS domain, the in-verse scattering series is (Weglein and Stolt, 1992; Mat-son, 1997):

V = V1 + V2 + V3 + · · · , (1)

where the Vn are determined using

D = G0V1G0, (2)

G0V2G0 = −G0V1G0V1G0, (3)

...

The perturbation is given by V =

�V PP V PS

V SP V SS

�, the

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Direct non-linear three parameter 2D elastic inversion

(causal) Green’s operator by G0 =

�GP

0 0

0 GS0

�and the

data by D =

�DPP DPS

DSP DSS

�.

Linear inversion of 1D elastic medium

Writing Eq. (2) explicitly leads to four equations.Assuming source and receiver depths are zero, in the(ks, zs; kg , zg; ω) domain, four equations relating the lin-ear components of the three elastic parameters and thefour data types:eDPP (kg, 0;−kg, 0; ω) = −

1

4

�1 −

k2g

ν2g

�ea(1)ρ (−2νg)

−

1

4

�1 +

k2g

ν2g

�ea(1)γ (−2νg) +

2k2gβ2

0

(ν2g + k2

g)α20

ea(1)µ (−2νg),

(4)eDPS(νg, ηg) = −

1

4

�kg

νg

+kg

ηg

�ea(1)ρ (−νg − ηg)

−

β20

2ω2kg(νg + ηg)

�1 −

k2g

νgηg

�ea(1)µ (−νg − ηg), (5)eDSP (νg, ηg) =

1

4

�kg

νg

+kg

ηg

�ea(1)ρ (−νg − ηg)

+β2

0

2ω2kg(νg + ηg)

�1 −

k2g

νgηg

�ea(1)µ (−νg − ηg), (6)

and eDSS(kg, ηg) = −

1

4

�1 −

k2g

η2g

�ea(1)ρ (−2ηg)

−

�η2

g + k2g

4η2g

−

2k2g

η2g + k2

g

�ea(1)µ (−2ηg), (7)

where ν2g + k2

g = ω2

α2

0

, η2g + k2

g = ω2

β2

0

, and aρ ≡ρ

ρ0− 1,

aγ ≡γ

γ0− 1, and aµ ≡

µ

µ0− 1 are the three parameters

we chose to do the elastic inversion.

Non-linear inversion of 1D elastic medium

Writing Eq. (3) in matrix form leads to the following fourequations

GP0 V PP

2 GP0 = −GP

0 V PP1 GP

0 V PP1 GP

0 − GP0 V PS

1 GS0 V SP

1 GP0 ,

GP0 V PS

2 GS0 = −GP

0 V PP1 GP

0 V PS1 GS

0 − GP0 V PS

1 GS0 V SS

1 GS0 ,

GS0 V SP

2 GP0 = −GS

0 V SP1 GP

0 V PP1 GP

0 − GS0 V SS

1 GS0 V SP

1 GP0 ,

GS0 V SS

2 GS0 = −GS

0 V SP1 GP

0 V PS1 GS

0 − GS0 V SS

1 GS0 V SS

1 GS0 .(8)

Since V PP1 relates to DPP , V PS

1 relates to DPS , and soon, the four components of the data will be coupled in

the non-linear elastic inversion. We cannot perform thedirect non-linear inversion without knowing all compo-nents of the data. As shown above, when we extend pre-vious work on two parameter acoustic case to the threeparameter elastic case, it is not just simply adding onemore parameter; there are more issues involved. Even forthe linear case, the solutions found in Eqs. (4) ∼ (7),are much more complicated than those of the acousticcase. For instance, four different sets of linear parame-ter estimates are produced from each component of thedata. Also, three or four distinct reflector mis-locationsarise from the two reference velocities (P-velocity and S-velocity). A particular non-linear approach has been cho-sen to side-step a portion of this complexity and addressour typical lack of four components of elastic data: weuse DPP as our fundamental data input, and perform areduced form of non-linear elastic inversion, concurrentlyasking: what beyond-linear value does this simpler frame-

work add? When assuming only DPP are available, first,

we compute the linear solution for a(1)ρ , a

(1)γ and a

(1)µ from

Eq. (4). Then, substituting the solution into the otherthree Eqs. (5), (6) and (7), we synthesize the other com-

ponents of data – DPS , DSP and DSS. Finally, using

the given DPP and the synthesized data, we perform thenon-linear elastic inversion, getting the following secondorder (first term beyond linear) elastic inversion solutionfrom Eq. (8),�1 − tan2 θ

�a(2)

ρ (z) +�1 + tan2 θ

�a(2)

γ (z) − 8b2 sin2 θa(2)µ (z)

= −

1

2

�tan4 θ − 1

� ha(1)

γ (z)i2

+tan2 θ

cos2 θa(1)

γ (z)a(1)ρ (z)

+1

2

��1 − tan4 θ

�−

2

C + 1

�1

C

��α2

0

β20

− 1

�tan2 θ

cos2 θ

�×

ha(1)

ρ (z)i2

− 4b2

�tan2 θ −

2

C + 1

�1

2C

��α2

0

β20

− 1

�tan4 θ

�× a(1)

ρ (z)a(1)µ (z)

+ 2b4

�tan2 θ −

α20

β20

��2 sin2 θ −

2

C + 1

1

C

�α2

0

β20

− 1

�tan2 θ

�×

ha(1)

µ (z)i2

−

1

2

�1

cos4 θ

�a(1)′

γ (z)

Z z

0

dz′

ha(1)

γ

�z′�− a(1)

ρ

�z′�i

−

1

2

�1 − tan4 θ

�a(1)′

ρ (z)

Z z

0

dz′

ha(1)

γ

�z′�− a(1)

ρ

�z′�i

+ 4b2 tan2 θa(1)′µ (z)

Z z

0

dz′

ha(1)

γ

�z′�− a(1)

ρ

�z′�i

+2

C + 1

1

C

�α2

0

β20

− 1

�tan2 θ

�tan2 θ − C

�b2

×

Z z

0

dz′a(1)µ z

�(C − 1) z′ + 2z

(C + 1)

�a(1)

ρ

�z′�

−

2

C + 1

2

C

�α2

0

β20

− 1

�tan2 θ

�tan2 θ −

α20

β20

�b4

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Direct non-linear three parameter 2D elastic inversion

×

Z z

0

dz′a(1)µ z

�(C − 1) z′ + 2z

(C + 1)

�a(1)

µ

�z′�

+2

C + 1

1

C

�α2

0

β20

− 1

�tan2 θ

�tan2 θ + C

�b2

×

Z z

0

dz′a(1)µ

�z′�a(1)

ρ z

�(C − 1)z′ + 2z

(C + 1)

�−

2

C + 1

1

2C

�α2

0

β20

− 1

�tan2 θ

�tan2 θ + 1

�×

Z z

0

dz′a(1)ρ

�z′�a(1)

ρ z

�(C − 1) z′ + 2z

(C + 1)

�, (9)

where a(1)ρ z

�(C−1)z′+2z

(C+1)

�= d

ha(1)ρ

�(C−1)z′+2z

(C+1)

�i/dz,

b = β0

α0and C =

ηg

νg. The first five terms on the right

side of Eq. (9) are inversion terms, i.e., they contributeto amplitude correction. The other terms on the rightside of the equation are imaging terms. Both the inver-sion terms and the imaging terms (especially the imagingterms) become much more complicated with the exten-sion to elastic media from acoustic (Zhang and Weglein,2005). The integrand of the first three integral terms isthe first order approximation of the relative change in P-

wave velocity. The derivatives a(1)′γ , a

(1)′ρ and a

(1)′µ in front

of those integrals are acting to correct the wrong locationscaused by the inaccurate reference P-wave velocity. Theother four terms with integrals will be zero as β0 → 0since in this case C → ∞. In the following, we test thisapproach numerically.

Numerical tests

For a single interface 1D elastic medium case, as shownin Fig. 1, the reflection coefficient RPP may be expressedanalytically (Foster et al., 1997). With this coefficient,similarly to the acoustic case, data may be expressed an-alytically (Clayton and Stolt, 1981; Weglein et al., 1997)as: eDPP (νg, θ) = RPP (θ)

e2iνga

4πiνg

, (10)

where a is the depth of the interface. Substituting Eq.(10)into Eq.(4), using k2

g/ν2g = tan2 θ and k2

g/(ν2g + k2

g) =

sin2 θ, Fourier transforming Eq.(4) over 2νg, and fixingdepth z > a and θ, we have

(1 − tan2 θ)a(1)ρ (z) + (1 + tan2 θ)a(1)

γ (z) − 8β2

0

α20

sin2 θa(1)µ (z)

= 4RPP (θ)H(z − a). (11)

In this section, we numerically test the direct inver-sion approach on the following model: shale (0.20porosity) over oil sand (0.30 porosity) with ρ0 =2.32g/cm3, ρ1 = 2.08g/cm3; α0 = 2627m/s, α1 =2330m/s; β0 = 1245m/s, β1 = 1488m/s. This high poros-ity model (30%) is typical of a weakly consolidated, shal-low reservoir sand. Pore fluids have a large impact on theseismic response. Density, P-wave velocity, and the α/β

q

111,, rba

000,, rba

PPT

PPR

SPR

SPT

Incident P-wave

Fig. 1: Response of incident compressional wave on a planarelastic interface. α0, β0 and ρ0 are the compressional wavevelocity, shear wave velocity and density of the upper layer,respectively; α1, β1 and ρ1 denote the compressional wave ve-locity, shear wave velocity and density of the lower layer. RPP ,RSP , T PP and T SP denote the coefficients of the reflectedcompressional wave, the reflected shear wave, the transmittedcompressional wave and the transmitted shear wave, respec-tively.

ratio of the oil sand are lower than the density, P-wave ve-locity, and α/β ratio of the overlying shale. Consequently,there is a significant decrease in density and P-bulk modu-lus and an increase in shear modulus at the shale/oil sandinterface. Using this model, we can find the correspond-ing RPP from the Zoeppritz equations. Then, choosingthree different angles θ1, θ2 and θ3, we can get the linear

solutions for a(1)ρ , a

(1)γ and a

(1)µ from Eq. (11), and then

get the solutions for a(2)ρ , a

(2)γ and a

(2)µ from Eq. (9).

There are two plots in each figure. The left ones are theresults for the first order, while the right ones are the re-sults for the first order plus the second order. The redlines denote the corresponding actual values. In the fig-ures, we illustrate the results corresponding to differentsets of angles θ1 and θ2. The third angle θ3 is fixed atzero. The numerical results indicate that all the second

010

2030

4050

6070

-0.22

-0.21

-0.20

-0.19

-0.18

-0.17

-0.16

-0.15

-0.14

-0.13

-0.12

-0.11

-0.10

0

1020

3040

5060

70

arh

o1+arh

o2

thet

a2

theta1

010

2030

4050

6070

-0.22

-0.21

-0.20

-0.19

-0.18

-0.17

-0.16

-0.15

-0.14

-0.13

-0.12

-0.11

-0.10

0

10

2030

4050

6070

arh

o1

thet

a2

theta1

Fig. 2: Shale (0.20 porosity) over oil sand (0.30 porosity).ρ0 = 2.32g/cm3, ρ1 = 2.08g/cm3;α0 = 2627m/s, α1 =2330m/s; β0 = 1245m/s, β1 = 1488m/s. For this model, the

exact value of aρ is -0.103. The linear approximation a(1)ρ (left)

and the sum of linear and first non-linear a(1)ρ + a

(2)ρ (right).

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Direct non-linear three parameter 2D elastic inversion

010

2030

4050

6070

-0.34

-0.33

-0.32

-0.31

-0.30

-0.29

-0.28

-0.27

-0.26

-0.25

-0.24

-0.23

-0.22

0

1020

3040

5060

70

agam

ma1+agam

ma2

thet

a2

theta1

010

2030

4050

6070

-0.34

-0.33

-0.32

-0.31

-0.30

-0.29

-0.28

-0.27

-0.26

-0.25

-0.24

-0.23

-0.22

0

10

2030

4050

6070

agam

ma1

thet

a2

theta1

Fig. 3: Shale (0.20 porosity) over oil sand (0.30 porosity).ρ0 = 2.32g/cm3, ρ1 = 2.08g/cm3;α0 = 2627m/s, α1 =2330m/s; β0 = 1245m/s, β1 = 1488m/s. For this model, the

exact value of aγ is -0.295. The linear approximation a(1)γ (left)

and the sum of linear and first non-linear a(1)γ + a

(2)γ (right).

010

2030

4050

6070

0.260.280.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.52

0.54

0

10

2030

4050

6070

am

u1

thet

a2

theta1

010

2030

4050

6070

0.260.280.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.52

0.54

0

10

2030

4050

6070

am

u1+am

u2

thet

a2

theta1

Fig. 4: Shale (0.20 porosity) over oil sand (0.30 porosity).ρ0 = 2.32g/cm3, ρ1 = 2.08g/cm3;α0 = 2627m/s, α1 =2330m/s; β0 = 1245m/s, β1 = 1488m/s. For this model, the

exact value of aµ is 0.281. The linear approximation a(1)µ (left)

and the sum of linear and first non-linear a(1)µ + a

(2)µ (right).

order solutions provide improvements over the linear forall of the parameters. When the second term is added tolinear order, the results become much closer to the cor-responding exact values and the surfaces become flatterin a larger range of angles. We interpret this as evidence(1) of the efficacy of the direct non-linear inversion equa-tions, and (2) that the data synthesis step is an acceptableapproximation.

Conclusion

Elastic non-linear direct inversion in 2D requires all fourcomponents of data. In this paper we present the firstdirect non-linear elastic equations, and analyze an algo-

rithm which requires only DPP and approximately syn-thesizes the other required components. Value-added re-

sults are obtained. Although DPP can itself provide use-ful non-linear direct inversion results, the implication ofthis research is that further value would derive from actu-ally measuring DPP , DPS , DSP and DSS, as the methodrequires.

Acknowledgements

The M-OSRP sponsors are thanked for supporting this

research. We are grateful to Robert Keys and DouglasFoster for useful comments and suggestions.

References

Foster D J, Keys R G and Schmitt D P 1997 Detectingsubsurface hydrocarbons with elastic wavefields, InInverse Problems in Wave Propagation, Volume 90of The IMA Volumes in Mathematics and its Appli-cations, 195–218. Springer.

Clayton R W and Stolt R H 1981 A Born-WKBJ inversionmethod for acoustic reflection data for attenuatingmultiples in seismic reflection data Geophysics 46

1559–1567

Innanen K A 2003 “Methods for the Treatment ofAcoustic and Absorptive/Dispersive Wave FieldMeasurements.” Ph.D. Thesis, University of BritishColumbia.

Liu F, Weglein A B, Innanen K A and Nita B G 2005 Ex-tension of the non-linear depth imaging capability ofthe inverse scattering series to multidimensional me-dia: strategies and numerical results 9th Ann. Cong.SBGf, Expanded Abstracts

Matson K H 1997 “An inverse scattering series method forattenuating elastic multiples from multicomponentland and ocean bottom seismic data.” Ph.D. Thesis,University of British Columbia. p 18

Shaw S A 2005 “An inverse scattering series algorithmfor depth imaging of reflection data from a layeredacoustic medium with an unknown velocity model.”Ph.D. Thesis, University of Houston

Weglein A B and Stolt R H 1992 Notes on approaches onlinear and non-linear migration-inversion, PersonalCommunication

Weglein A B, Gasparotto F A, Carvalho P M and Stolt RH 1997 An inverse-scattering series method for atten-uating multiples in seismic reflection data Geophysics62 1975–1989

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 R27–R83

Zhang H and Weglein A B 2005 The inverse scattering se-ries for tasks associated with primaries: Depth imag-ing and direct non-linear inversion of 1D variable ve-locity and density acoustic media 75th Ann. Inter-nat. Mtg., Soc. Expl., Geophys., Expanded Abstracts1705–1708

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EDITED REFERENCESNote: This reference list is a copy-edited version of the reference list submitted by theauthor. Reference lists for the 2006 SEG Technical Program Expanded Abstracts havebeen copy edited so that references provided with the online metadata for each paper willachieve a high degree of linking to cited sources that appear on the Web.

REFERENCES 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, 1559–1567.

Foster, D. J., R. G. Keys, and D. P. Schmitt, 1997, Detecting subsurface hydrocarbons with elastic wavefields: Inverse Problems in Wave Propagation, 90, 195–218.

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

Liu, F., A. B. Weglein, K. A. Innanen, and B. G. Nita, 2005, Extension of the non-linear depth imaging capability of the inverse scattering series to multidimensional media: Strategies and numerical results: 9th Annual Conference, SBGf.

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.

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.

Weglein, A. B., F. V. Araújo, 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 Problem, 19, R27–R83.

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Zhang, H., and A. B. Weglein, 2005, The inverse scattering series for tasks associated with primaries: Depth imaging and direct non-linear inversion of 1D variable velocity and density acoustic media: 75th Annual International Meeting, SEG, Expanded Abstracts, 1705–1708.

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