Perturbation methods for two special cases of the time-lapse seismic inverse problem

Geophysical Prospecting, 2014, 62, 453–474 doi: 10.1111/1365-2478.12105

Perturbation methods for two special cases of the time-lapse seismicinverse problem

Kristopher A. Innanen1, Mostafa Naghizadeh2 and Sam T. Kaplan3

1Dept. of Geoscience, University of Calgary, Calgary, AB, 2Dept. of Geoscience, University of Calgary, Calgary, AB (currently Shell Canada,Calgary, AB), and 3ConocoPhillips, Houston, TX (Currently, Chevron Energy Technology, Houston, TX)

Received May 2012, revision accepted January 2013

ABSTRACTScattering theory, a form of perturbation theory, is a framework from within whichtime-lapse seismic reflection methods can be derived and understood. It leads to ex-pressions relating baseline and monitoring data and Earth properties, focusing ondifferences between these quantities as it does so. The baseline medium is, in thelanguage of scattering theory, the reference medium and the monitoring mediumis the perturbed medium. The general scattering relationship between monitoringdata, baseline data, and time-lapse Earth property changes is likely too complexto be tractable. However, there are special cases that can be analysed for physicalinsight. Two of these cases coincide with recognizable areas of applied reflection seis-mology: amplitude versus offset modelling/inversion, and imaging. The main resultof this paper is a demonstration that time-lapse difference amplitude versus offsetmodelling, and time-lapse difference data imaging, emerge from a single theoreticalframework. The time-lapse amplitude versus offset case is considered first. We con-strain the general time-lapse scattering problem to correspond with a single immobileinterface that separates a static overburden from a target medium whose propertiesundergo time-lapse changes. The scattering solutions contain difference-amplitudeversus offset expressions that (although presently acoustic) resemble the expressionsof Landro (2001). In addition, however, they contain non-linear corrective termswhose importance becomes significant as the contrasts across the interface grow. Thedifference-amplitude versus offset case is exemplified with two parameter acoustic(bulk modulus and density) and anacoustic (P-wave velocity and quality factor Q)examples. The time-lapse difference data imaging case is considered next. Instead ofconstraining the structure of the Earth volume as in the amplitude versus offset case,we instead make a small-contrast assumption, namely that the time-lapse variationsare small enough that we may disregard contributions from beyond first order. Aninitial analysis, in which the case of a single mobile boundary is examined in 1D, jus-tifies the use of a particular imaging algorithm applied directly to difference data shotrecords. This algorithm, a least-squares, shot-profile imaging method, is additionallycapable of supporting a range of regularization techniques. Synthetic examples verifythe applicability of linearized imaging methods of the difference image formationunder ideal conditions.

Key words: Perturbation, Time lapse, Scattering.

∗E-mail: [email protected]

INTRODUCTION

In this paper we investigate the character of time-lapse seis-mic reflection methods derivable from a self-consistent per-turbation approach. The results, though at present restricted

C© 2014 European Association of Geoscientists & Engineers 453

454 K.A. Innanen, M. Naghizadeh and S.T. Kaplan

to acoustic media, are suggestive of some modest correctiveterms to be added to existing time-lapse amplitude methods incases of large contrasts. Furthermore, they permit us to viewtime-lapse imaging and AVO-inversion methods as being de-rived from within a common framework.

In time-lapse, or 4D, seismic monitoring (Greaves andFulp 1987; Lumley 2001; Denli and Huang 2009), the stan-dard seismic survey is augmented with a calendar time axis.Data variations along this axis provide a means to character-ize monthly, yearly, or decade-length evolution of a particularvolume of Earth. Originally developed for monitoring of hy-drocarbon production, the applications of time-lapse seismicappear to be expanding, in particular in the area of carbonstorage. In this application time-lapse is expected to be a keytechnology both in the short term (i.e., in monitoring injectiv-ity) and the long term (i.e., in monitoring storage fidelity).

From a technical point of view, development of methodsfor:1. Analysis and inversion of differenced reflection ampli-tudes, i.e., time-lapse AVO signatures, to determine targetproperty changes (Landro 2001);2. Imaging of difference data to characterize structuralchanges (Ayeni and Biondi 2010); and3. Practical means to cope with preprocessing issues particu-lar to the time-lapse problem, such as redatuming (Winthae-gen, Verschuur, and Berkhout 2004), image registration(Fomel and Jin 2009), and discrimination of target versusacquisition variations (Berkhout and Verschuur 2007);represent open and active areas of research. Other researchareas, for instance time-lapse geomechanics (Minkoff et al.

2004), also impact seismic methods, but in this paper we willfocus on the above list.

Perturbation (or scattering) theory is a framework thatgives rise to a wide range of non time-lapse seismic processingand inversion methods (e.g., Weglein et al. 2003). The idea ofapplying scattering-type perturbation methods to time-lapseproblems of types 1–3 above has been proposed in the pastand applied with significant success to synthetic data (Zhang2006).

However, to date, scattering theory has been applied totime-lapse problems inconsistently. Formulations involvingdifferences between reference (i.e., baseline) reflections andperturbed (i.e., monitoring) reflections have been derived as-suming homogeneous reference media. The inconsistency liesin the inability of a homogeneous reference medium to causethe baseline reflection. We will instead treat the problem withreference media chosen to describe complex, heterogeneousbaseline Earth volumes, capable of causing and/or altering a

significant number of the events in a differenced data set. Ourpurpose in this paper is to investigate the consequences ofsuch an application, and the possibility of deriving practicalprocessing algorithms based thereon.

Not all of the consequences are positive. In perturbationtheory the reference medium is assumed to be known exactly,as is the wavefield propagating through that medium. Sincethere is no way to achieve that precise state of knowledge for acomplex reference Earth volume, arriving at practical methodsthis way is far from guaranteed. Philosophically similar con-cerns with the full time-lapse inverse problem have led to thedevelopment of alternative techniques for 4D analysis, suchas warping (Williamson, Cherrett, and Sexton 2007; Grandietet al. 2009). Our approach (see also Innanen and Naghizadeh2011; Innanen, Naghizadeh, and Kaplan 2011) is to remainin the full-wave perturbation environment but therein iden-tify two special cases, that (i) permit a self-consistent per-turbation approach to proceed, assuming only a reasonableamount of information about the baseline/reference mediumand (ii) correspond to important areas of technology develop-ment, specifically, items (1) and (2) in the earlier list.

The paper is organized such that the two special cases arediscussed in turn. In the first section we show that a full-waveperturbation treatment of the time-lapse problem reduces toa difference-AVO problem when the perturbation is fixed tobe a single immobile interface. This leads to a prototype for-mulation of the time-lapse amplitude difference problem thatis then applied to two multi-parameter acoustic/anacousticconfigurations. The basic linear and non-linear forward andinverse problems are both formulated. In the second sectionwe treat the structural variation problem, assuming small con-trasts. We show, again using a simple (but full-wave) perturba-tion model, why imaging methods based on non time-lapse as-sumptions are appropriate for application to differenced data,in spite of an apparent inconsistency. Thereafter we examinenumerically the use of a least-squares shot-profile implemen-tation of linearized time-lapse inversion. Our conclusions re-gard potentially useful directions to continue development ofboth of these nascent formulations.

Summary of terms

A scattering description involves the definition of a referencemedium and a perturbed medium. The difference betweenthese two media is represented by a ‘perturbation’, and thecorresponding difference between the wavefields in those me-dia is the ‘scattered field’. The perturbation and scattered fieldare related non-linearly through the Born series and if the

C© 2014 European Association of Geoscientists & Engineers, Geophysical Prospecting, 62, 453–474

Perturbation methods for time-lapse seismic inversion 455

perturbation is small in magnitude and occupies only a smallspatial interval, this relationship may be linearized, formingthe Born approximation. Meanwhile in the time-lapse surveythere are several data sets, a baseline data set and one or moremonitoring data sets. The baseline data set subtracted fromone of the monitoring data sets form the ‘difference data’, andthe totality of all changes in the properties of the target volumethat have accrued in the interim between surveys is the ‘differ-ence model’. To apply a scattering theoretic framework to adescription of time-lapse seismic reflection data is to associatethe reference model with the baseline medium, the referencewavefield with the baseline data, the perturbed model withthe monitoring medium, and the perturbed wavefield with themonitoring data. The difference data are, in this framework,equal to the scattered field evaluated on the measurement sur-face. Table 1 contains a summary of the symbols used in thispaper and their meaning.

A M P L I T U D E V A R I A T I O N S A T A NIMMOBILE INT E R FA C E

Summary

Consider the time-lapse scenario illustrated in Fig. 1, in whichan immobile interface separates a static overburden from atarget medium with time-varying acoustic properties (cor-responding to, e.g., caprock overlying a reservoir rock dur-ing production, or CO2 injection). The dominant time-lapsesignature in this scenario is a variation in the amplitude-variation-with-offset or AVO response of the interface. Lan-dro (2001) derived expressions for the time-lapse change in theangle-dependent elastic reflection strengths of such interfaces.In this section we will examine the extent to which acous-tic and anacoustic versions of such formulas may be deriveddirectly from a perturbative treatment of the full-wave time-lapse problem. The results are modelling/inversion formulaswith both linear terms and various types of second-order cor-rection. After demonstrating that these full-wave results canbe reproduced from an analysis of the difference reflectioncoefficient in isolation (a much less laborious procedure), weextend our results to multi-parameter contrasts and considertheir numeric behaviour.

Initial analysis

Scattering or perturbation theory applied to wave problems re-quires a reference wave equation and a perturbed wave equa-tion to be selected. Our initial analysis involves a 1D scalarenvironment, in which the reference and perturbed equations

are selected to represent an immobile target with a chang-ing wave velocity (for background information on how to setup and solve such series expansions, see e.g., Matson 1996;Weglein et al. 2003). The results become the prototype foramplitude or AVO difference modelling in time-lapse seismicmonitoring.

Scattering from an amplitude-perturbed interface

In the frequency ω domain, the reference or baseline mediumand wavefield are defined such that[

d2

dz2+ ω2

c2BL(z)

]G(z, zs) = δ(z − zs), where

1c2

BL(z)=

{c−2

BL , z > z1

c−20 , z < z1

, (1)

where c0 is the static incidence medium velocity and cBL isthe target velocity at the time of the baseline survey. Thedepth z represents the observation depth, and zs represents thesource depth.The target interface is at depth z1. The perturbedmedium and wavefield are defined by[

d2

dz2+ ω2

c2M(z)

]P(z, zs) = δ(z − zs), where

1c2

M(z)=

{c−2

M , z > z1

c−20 , z < z1

, (2)

where the target velocity has evolved into cM. The interfacedepth remains fixed at z1. The perturbed field P is next ex-panded in orders of a perturbation that measures the differ-ence between cBL and cM. To model the changing velocityof the single reflector, we define a dimensionless time-lapseperturbation

ac(z) = ac H(z − z1), (3)

where H is the Heaviside or step function and where ac =1 − c2

BL/c2M. In Appendix A we calculate the first three orders

of this expansion, i.e., we form the leading order terms of theBorn series

P(zg, zs) = P (0)(zg, zs) + P (1)(zg, zs) + P (2)(zg, zs) + · · · . (4)

Our interest is to understand what this series can tell usabout the relationship between the difference model, which inour case is represented by ac(z) and the difference data, whichare the differences between monitoring and baseline data. Thelatter quantities, in this context, are related to �P = P − P (0),since P (0) = G. Setting zg = zs = 0 and noting that k0 = ω/c0,



(a) (b)

θ θ

RBL RM

static overburden static overburden

baseline target monitoring target

Figure 1 Time-lapse response of an immobile interface. (a) A static overburden, overlying a baseline target whose properties will vary on thecalendar time scale, gives rise to a reflection strength RBL(θ ). (b) The same volume of Earth at the time of the monitoring survey, when the targetproperties and the reflection strength RM(θ ) have both changed.

the remainder of the terms in the series in equation (4), ascalculated in Appendix A, provide us with an expression for�P:

�P =[

14

ac

(1 − R2

BL

)

+ 18

a2c

(1 − 1

2RBL − R2

BL

)+ · · ·

]ei2k0z1

i2k0, (5)

where RBL is the reflection coefficient caused by the interfaceat the time of the baseline survey:

RBL = cBL − c0

cBL + c0. (6)

The phase of �P is determined by the rightmost element ofequation (5). Because the location of the interface is fixed atz1, the phase term is unchanging, and appears as a commonfactor in all of the series expansion terms. The only substantialeffect of the expansion is to express the change in amplitudefrom the baseline to the monitoring surveys. This is the seriesin brackets [·] in equation (5). Our interpretation is, in otherwords, that the full perturbative time-lapse expansion ulti-mately reduces to the computation of �R = RM − RBL, andthat this takes the form of the series

�R = 14

ac

(1 − R2

BL

) + 18

a2c

(1 − 1

2RBL − R2

BL + · · ·)

+ · · · .

(7)

A self-consistent perturbative model of difference amplitudes,therefore, appears as a series, in terms of both the time-lapsemedium perturbation ac and baseline wave quantities RBL.This series is the time-lapse forward problem, i.e., an expres-

sion of the difference data in terms of the baseline quantitiesand the time-lapse changes from cBL to cM. If the problem hadbeen posed inconsistently, i.e., the baseline/reference mediumhad not been treated as a medium with the capacity to reflect,but had been assigned a homogeneous velocity c0, the result-ing expression for �R would be as in equation (7) with RBL

set to 0.In Fig. 2 we illustrate the importance of the leading ver-

sus higher order terms in the expansion in equation (7), defin-ing the order of a term aN

c × RMBL to be N + M. The percent-

age error, 100% × (�Rexact − �Rapprox)/�Rexact, is plotted fora range of cBL and cM values in three of the panels, witheach panel representing a different order of approximation. InFig. 2(a), the exact difference �Rexact is plotted. In Fig. 2(b),equation (7) is linearized and the panel illustrates the first-order error; the approximation is truncated at second orderin Fig. 2(c), and at third order in Fig. 2(d). The convergenceover a reasonable range of baseline and monitoring target ve-locities is observed to be relatively rapid. Especially for largecontrasts the second-order and third-order terms can have asignificant influence, where that influence includes both cou-pling between baseline quantities (RBL) and time-lapse pertur-bations (ac). In this paper we will typically consider second-order corrections.

Direct expansion of the difference reflection coefficient

Part of the value of the expansion in equation (7) is thatit is traceable to a full perturbative treatment of the time-lapse problem, beginning with the basic wave equations.



Figure 2 Error plots for approximations of �R. Panel (a) is a plot of the exact �R for each baseline and monitoring target velocity pair. Inthe remainder of the panels, (�Rexact − �Rapprox)/�Rexact × 100% is plotted for the same range of cBL and cM values. Each panel is a differentapproximation: (a) exact, (b) first order, (c) second order, (d) third order.

However, having derived it, it would be convenient to de-velop more direct means of reproducing the terms, as expan-sions involving waves at oblique incidence, and media withmultiple parameters involve significantly more sophisticatedcomputation.

We proceed by again defining a time-lapse perturba-tion ac = 1 − c2

BL/c2M, but additionally introducing a baseline

perturbation bc = 1 − c20/c2

BL, which measures the size of thecontrast causing the baseline reflection RBL. Thus two ratiosof the time-lapse velocities may be formed:

c0

cBL= (1 − bc)

1/2,cBL

cM= (1 − ac)

1/2. (8)

We turn next to the difference reflection coefficient�R = RM − RBL, paying particular attention to the monitor-

ing coefficient:

�R = cM − c0

cM + c0− RBL

=1 −

(c0cBL

) (cBLcM

)1 +

(c0cBL

) (cBLcM

) − RBL, (9)

into which we may substitute the forms in equation (8). Ex-panding these as binomial series, we obtain

�R = �

1 + �− RBL, (10)

where

� = 14

ac + 14

bc + 116

a2c − 1

8bcac + 1

16b2

c + · · · . (11)



We next consider the baseline reflection alone. It may be ex-panded in series about bc:

RBL = cBL − c0

cBL + c0

= 14

bc + 18

b2c + · · · , (12)

and this simpler series can be reverted (Abramowitz and Ste-gun 1972, pg. 16) such that the inverse series solution for bc

is determined:

bBL = 4RBL − 8R2BL + · · · . (13)

We substitute equation (13) into equation (11) and this in turnis substituted into equation (10). The denominator (1 + �)−1

in equation (10) may then be expanded in a final binomialseries, resulting in

�R = 14

ac

(1 − R2

BL

) + 18

a2c

(1 − 1

2RBL − R2

BL + · · ·)

+ · · · ,

(14)

which is the same as the series found in equation (5). Theresults of a full, self-consistent perturbative treatment of thetime-lapse AVO problem can be reproduced through directanalysis of the difference reflection coefficient, leading to aconsiderable savings in calculation. We will use the procedurein equations (8)–(14) as a prototype for more complex cases.

Scalar inversion at normal incidence

In equation (13) we employed a series reversion to transform afunction of bc into a function of RBL, obtaining equation (14).This reversion is, in essence, an inverse procedure and we mayadapt it to examine the problem of determining the time-lapseperturbation ac from measurements of �R. We begin withthe original series expansion of the measured quantity �R inequation (14). Let us consider the baseline quantity RBL to beknown, and the time-lapse quantity ac to be unknown. Weexpand the unknown perturbation in the series

ac = ac1+ ac2

+ · · · , (15)

where acnis defined to be nth order in �R. Substituting equa-

tion (15) into equation (14) and equating like orders, a se-quence of equations allowing acn

to be solved is obtained.Summing these, the following inverse series is obtained:

ac = 4�R(

11 − R2

BL

)

− 8�R2

[1 − (1/2)RBL − R2

BL

(1 − R2BL)3

]+ · · · . (16)

This is a non-linear formula for the reconstruction of thetime-lapse change in the target velocity, ac. It is the inverseof the series in equation (14). The reconstruction relies onboth time-lapse difference data (�R) and baseline data (RBL)as input; the two couple to produce the correct answer.

To illustrate the importance of including baseline infor-mation (RBL) as well as difference information (�R), we il-lustrate the action of equation (16) numerically, both withRBL included correctly, and neglected. In Figs 3 and 4 we il-lustrate these two cases numerically for ranges of monitoringtarget velocity values cM, given a fixed incidence medium ve-locity (c0 = 1500 m/s) and baseline target velocity (in Fig. 3,cBL = 3950 m/s, and in Fig. 4, cBL = 3150 m/s). The exact ve-locity is plotted on the vertical axis, and the recovered velocityon the horizontal axis; the line representing perfect recoveryis plotted in bold on each panel. In Fig. 3(a,b), the solid linerepresents the reconstructed monitoring velocity incorporat-ing RBL, and the dashed line represents the reconstructionneglecting it, both using the first term in equation (16) only.In Fig. 3(c,d), the same results are plotted using both terms inequation (16). Fig. 3(a,b) differs in that it focuses on differentranges of the cM value. Figure 4 is organized in the same way.

We conclude that there is a persistent and significant in-crease in the inversion accuracy when both terms in equation(16) are incorporated and a concurrent increase in accuracywhen the baseline wave quantities are correctly incorporatedalso. In every case the highest accuracies are obtained usingthe full two terms in equation (16); in larger contrast casesthird and higher order terms are expected to also contributemeaningfully.

Case 1: Time-lapse variations in target density and bulkmodulus

We next extend the 1D, normal incidence, scalar results of theprevious section to include multiple parameter, oblique inci-dence configurations. The results here are acoustic (and lateranacoustic); full elastic forms will be reported on in futurecommunications.

Expansion in terms of time-lapse and baseline perturbations

We begin with exact expressions for acoustic reflection coeffi-cients appropriate for baseline and monitoring target half-spaces with acoustic properties κBL, ρBL and κM, ρM andincidence-medium properties κ0, ρ0:

RBL(θ ) = 1 − �BL(θ )1 + �BL(θ )

, and RM(θ ) = 1 − �M(θ )1 + �M(θ )

, (17)



2000 2500 3000 35001500

2000

2500

3000

3500(a)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

3500 4000 45003000

3500

4000

4500

5000(b)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

2000 2500 3000 35001500

2000

2500

3000

3500(c)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

3500 4000 45003000

3500

4000

4500

5000(d)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

Figure 3 Reconstructions of cM plotted against exact cM. Bold line represents perfect reconstruction. Either the 1st or both the 1st and 2ndterms in equation (16) are used. (a) 1st term with RBL = 0 (dashed) versus 1st term with RBL included (solid), low cM range; (b) same as (a)but high cM range; (c) both terms with RBL = 0 (dashed) versus both terms with RBL included (solid); (d) same as (c) but high cM range. Fixedmedium properties c0 = 1500 m/s, cBL =3950 m/s.

where, using X to indicate BL or M,

�X(θ ) =(

ρ0

ρX

) (κ0

κX

ρX

ρ0

)1/2

(1 − sin2 θ )−1/2

(1 − κX

κ0

ρ0

ρXsin2 θ

)1/2

. (18)

Following the prototype problem in the previous section, webegin by introducing the four perturbations

bκ = 1 − κ0

κBL, bρ = 1 − ρ0

ρBL,

aκ = 1 − κBL

κM, aρ = 1 − ρBL

ρM, (19)

where the letter b again means perturbations across the bound-ary at the time of the baseline survey, and the letter a meanstime-lapse perturbations. In Appendix B we show how, af-ter substitution of these quantities into the explicit expressionfor the acoustic difference reflection coefficient, it may be ex-

panded such that a series for �R(θ ) = RM(θ ) − RBL(θ ) of

�R(θ ) = Aκ (θ )aκ + Aρ(θ )aρ + Aκκ (θ )a2κ

+ Aκρ(θ )aκbρ + Aρρ(θ )a2ρ + · · · , (20)

is obtained (the A factors are given in the Appendix). As inthe scalar case, the expansion in equation (20) contains termsin the time-lapse perturbations aκ and aρ both alone and alsocoupled with baseline perturbations bρ and bκ , the latter ap-pearing in the A factors.

The relative importance of the terms in equation (20)is examined in Fig. 5. In Fig. 5(a) an example pair of syn-thetic baseline (solid) and monitoring (dashed) reflection co-efficients is illustrated. The difference between these twocurves–the exact �R(θ ) curve–is plotted as a bold line inFig. 5(b). In Fig. 5(c), the exact �R is again plotted inbold, and the sum of the first two terms of equation (20) is



2000 2500 3000 35001500

2000

2500

3000

3500

(a)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

3500 4000 45003000

3500

4000

4500

5000(b)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

2000 2500 3000 35001500

2000

2500

3000

3500

(c)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

3500 4000 45003000

3500

4000

4500

5000(d)

Exact cM

(m/s)

Rec

over

ed c

M (

m/s

)

Figure 4 Reconstructions of cM plotted against exact cM. Bold line represents perfect reconstruction. Either the 1st or both the 1st and 2ndterms in equation (16) are used. (a) 1st term with RBL = 0 (dashed) versus 1st term with RBL included (solid), low cM range; (b) same as (a)but high cM range; (c) both terms with RBL = 0 (dashed) versus both terms with RBL included (solid); (d) same as (c) but high cM range. Fixedmedium properties c0 = 1500 m/s, cBL =3150 m/s.

plotted as a solid curve. The difference between these twocurves is suggestive of significant linearization error. To illus-trate the influence of the baseline terms in the A factors (i.e.,the baseline/time-lapse coupling), on the same plot the sameapproximation is plotted with bκ and bρ having been set tozero (dashed line).

In Fig. 5(d) we repeat the three plots, but this time us-ing all five of the terms in equation (20). The change inaccuracy between the solid curves in Figures 5(c,d) is an indi-cation of the importance of the second-order corrections whentime-lapse and/or baseline contrasts are large. A further, moresubtle, change in accuracy is noted between the solid curveand the dashed curve in Fig. 5(d), illustrating the uplift in ac-curacy coming from correct incorporation of the coupling oftime-lapse and baseline quantities.

The modelling of difference-AVO from a perturbationframework produces series expansions of the difference data,

which can be linearized when contrasts are small. The lin-earizations are acoustic versions of the elastic forms due toLandro (2001), with the quantities aκ and aρ playing the rolesof Landro’s elastic relative changes �VP/VP , �VS/VS and�ρ/ρ. We conclude that if both (either) the physical contrastsacross the boundary and (or) the time-lapse variations, arelarge, second and higher order corrections of the type deriv-able with the current approach are significant.

Acoustic inversion at oblique incidence

The formula in equation (20) is, essentially, a forward mod-elling equation for acoustic two-parameter difference am-plitudes, which now may be used as a driver for any suit-able inversion algorithm. For completeness, we will here in-clude a formal procedure for its direct inversion, of the typethat produced equation (16) in the scalar case. Since two



Figure 5 Approximating �R, case 1: acoustic two parameter example. (a) Input reflection coefficients: RBL (solid) versus RM dashed. (b) Exact�R = RM − RBL. (c) First-order �R approximations, with all baseline and time-lapse perturbations included (solid) and with bκ = bρ = 0(dashed). Exact �R is in bold. (d) second-order �R approximations, with all baseline and time-lapse perturbations included (solid) and withbκ = bρ = 0 (dashed). Exact �R is in bold. Model parameters: c0 = 1500 m/s, cBL = 2200 m/s, cM = 2000 m/s, ρ0 = 2.0 gm/cc, ρBL = 2.4gm/cc, ρM = 2.2 gm/cc. Input bulk moduli were mapped from these values using κ = c2ρ × 103.

parameters undergo time-lapse change, a reflection coeffi-cient datum at normal incidence is not sufficient to constrainthe estimation. We consider the use of several angles in thissection.

We set out two inverse series expansions, one for aκ

and one for aρ . These will be substituted into the forwardmodelling series in equation (20). In this paper, in order tostudy the time-lapse problem in isolation, we have takenthe view that all baseline quantities are known (i.e., havebeen accurately estimated externally to the current proce-dure), so we shall also consider the functions A to be given.We have

aκ = aκ1+ aκ2

+ · · ·aρ = aρ1

+ aρ2+ · · · , (21)

where aκnand aρn

are nth order in any measurements of �R(θ )to be used. These series are substituted into equation (20) andterms of equal order in a set of N data

�R1 = [�R(θ1),�R(θ2), . . . , �R(θN)]T (22)

are equated. At first order we obtain the equation

�R1 = A

[aκ1

aρ1

], (23)

where

A =

⎡⎢⎢⎢⎢⎣

Aκ (θ1) Aρ(θ1)Aκ (θ2) Aρ(θ2)

......

Aκ (θN) Aρ(θN)

⎤⎥⎥⎥⎥⎦ . (24)



Provided N > 1, this may be solved in any suitable manner,e.g., via least squares:[

aκ1

aρ1

]= (

ATA)−1

AT�R1, (25)

whereby the first-order terms for aκ and aρ are obtained. Wenext turn to the second-order components of the inverse series.These terms may be organized such that the same matrix A isinverted to determine aκ2

and aρ2:[

aκ2

aρ2

]= (

ATA)−1

AT�R2, (26)

but the vector that at first order contained the difference re-flection coefficients now contains second-order operations onaκ1

and aρ1:

�R2 = −

⎡⎢⎢⎢⎢⎣

Aκκ (θ1)a2κ1

+ Aκρ(θ1)aκ1aρ1

+ Aρρ(θ1)a2ρ1

Aκκ (θ2)a2κ1

+ Aκρ(θ2)aκ1aρ1

+ Aρρ(θ2)a2ρ1

...Aκκ (θN)a2

κ1+ Aκρ(θN)aκ1

aρ1+ Aρρ(θN)a2

ρ1

⎤⎥⎥⎥⎥⎦ . (27)

This may continue to any order desired.

Expansion in terms of time-lapse perturbations and baseline

reflection strengths

In our prototype expansion (equation 14) we undertook toexpress �R in terms of the time-lapse perturbation and base-line reflection coefficient; thus far we have expressed the multiparameter versions in terms of two perturbations. The impor-tance of expressing �R in terms of RBL lay in establishing thatthe full-wave perturbation results were correctly reproducedby a direct expansion technique. We may likewise transformequation (20). Consider the baseline reflection coefficient ex-panded in isolation:

RBL(θ ) = 14

(1 + sin2 θ

)bκ + 1

4

(1 − sin2 θ

)bρ + · · · . (28)

Since

∂ RBL(θ )∂θ

= 2(bκ − bρ) sin θ cos θ + · · · , (29)

and so consequently

bκ − bρ = 12 sin θ cos θ

∂ RBL(θ )∂θ

+ · · · , (30)

we find that the difference reflection coefficient in equation(20) can be expressed in terms of time-lapse perturbations andRBL(θ ). Using a subset of the terms derived in Appendix B, the

difference coefficient becomes

�R(θ ) = 14

(1 + sin2 θ

)aκ + 1

4

(1 − sin2 θ

)aρ

+ 18

tan θ∂ RBL(θ )

∂θ(aκ − aρ) + 1

8

(1 + 2 sin2 θ

)a2

κ

− 14

sin2 θaκaρ + 18

a2ρ + · · · . (31)

Case 2: Time-lapse variations in P-wave velocity and Q

As a further illustrative case, we consider a target with anelas-tic properties (i.e., P-wave velocity and quality factor Q),which vary during a time-lapse experiment. We assign to theincidence medium a scalar wave velocity c0. To the target weassign the velocities cBL and cM to correspond with the timesof the baseline and monitoring surveys respectively. Further,the target is assigned baseline and monitoring quality factorsQBL and QM respectively.

In Appendix C we show that, adopting a nearly-constantQ model (Aki and Richards 2002) for the target and express-ing the baseline and monitoring reflection coefficients as perInnanen (2011), the corresponding expansion of �R is

�R(θ, ω) = 14

(1 + sin2 θ

)ac +

[14

bc − 12

F (ω)bQ

]sin2 θ ac

+ 12

(1 + sin2 θ )F (ω)bQaQ

+ 18

(1 + 2 sin2 θ )a2c + · · · , (32)

where

bc = 1 − c20

c2BL

, bQ = 1QBL

,

ac = 1 − c2BL

c2M

, aQ = 1 − QBL

QM, (33)

and F (ω) = i/2 − (1/π ) log(ω/ω0). The parameter ω0 is a ref-erence frequency.

To evaluate the form in equation (32) as to its useful-ness as a driver for time-lapse inversion, we again comparethe relative importance of its first and second-order contribu-tions. The results are illustrated in Fig. 6. Since in anacous-tic problems the reflection strengths are generally functionsof frequency as well as angle, we repeat the same illustra-tion twice, once for a relatively high frequency (80 Hz inFig. 6 a-c) and one for a relatively low frequency (10 Hz inFig. 6 b-d). In Fig. 6(a,b) the real parts of RBL (as a solid line)and RM (as a dashed line) are illustrated. In Fig. 6(c,d) thecorresponding exact �R (as a bold, solid line) is comparedwith its first-order approximation (as a dashed line), which is



0 10 20 300

0.02

0.04

0.06

0.08

0.1

0.12

(a)

Re

R

(deg)0 10 20 30

0

0.02

0.04

0.06

0.08

0.1

0.12

(b)

Re

R

(deg)

0 10 20 300

0.01

0.02

0.03

0.04

0.05

0.06

Re

R

(c)

(deg)0 10 20 30

0

0.01

0.02

0.03

0.04

0.05

0.06

Re

R

(d)

(deg)

Figure 6 Approximating �R, case 2: anacoustic two parameter example. (a) Input reflection coefficients at frequency f = 10 Hz: RBL (solid)versus RM dashed. (b) Input reflection coefficients at frequency f = 80 Hz: RBL (solid) versus RM dashed. (c) �R calculations: exact (boldsolid), First-order approximate (dashed) and with second-order corrections (solid) atfrequency f = 10 Hz. (d) �R calculations: exact (boldsolid), First-order approximate (dashed) and with second-order corrections (solid) atfrequency f = 80 Hz. Model parameters: c0 = 2000 m/s,cBL = 2200 m/s, cM = 2400 m/s, QBL = 70, QM = 10.

formed from the first term in equation (32) and the combinedfirst- second-order terms (as a solid line).

The time-lapse change in Q does not contribute to �R

at first order. Nevertheless, its influence via the second-orderterms is evidently quite important. We again conclude that,except for small contrasts, second-order terms, both involvingtime-lapse and baseline quantities, must be expected to play animportant role in determining the difference reflection coeffi-cient, and, ultimately, the time-lapse inverse problem, thoughthis last statement is preliminary and can only be establishedwith field cases.

STRUCTURAL V A R I A T I ON S OF MOBI LEAND IMMOBILE FORMATIONS

Summary

Consider the time-lapse scenario in Fig. 7, in which two ex-amples of ‘structural’ time-lapse variations are illustrated. In

the first example (Fig. 7 a), the Earth volume is truly mo-bile, in the sense that the boundary of a propagating plumechanges in location from the time of the baseline survey tothe time of the monitoring survey. A plausible time-lapsegoal is to image the regions of the Earth model occupied bythe changing boundary. The second example (Fig. 7 b) in-volves a change in the scalar velocity of an otherwise fixedtarget. We nevertheless classify this as a structural prob-lem, since a plausible time-lapse goal is to image the re-gions in which change has occurred. In this section we willconsider perturbation methods for characterizing changes inan Earth volume of either type. As before, we begin with asimple 1D and scalar, but full-wave, analysis. We establishthe suitability of a linearized model, in spite of an appar-ent theoretical problem with this approximation and imple-ment a multidimensional imaging algorithm based on it. A2D synthetic example illustrates the basic behaviour of theapproach.



Figure 7 Illustration of two cases classifiable as involving time-lapse structural variation. (a) In which the volume structure undergoes a spatialvariation in transitioning from the top panel (baseline Earth volume) to the bottom panel (monitoring Earth volume). (b) In which the volumestructures remain spatially static but properties within a particular sub-volume vary.

A motivating example

We will make use of a linearized scattering model to formdifference images from difference data. The role of our ini-tial theoretical analysis is to provide a rationale for thatchoice.

The issue is well illustrated by an attempt to apply aninconsistent, ‘non time-lapse’, linearized inverse scatteringmethod (i.e., with a reference medium too simple to gener-ate important baseline events) to time-lapse difference data.In a 1D scalar acoustic setting, with a homogeneous referencemedium, such imaging reduces to trace integration (i.e., theimpedance inversion of Lindseth 1979), a fact established in

early studies of inverse scattering (e.g., Bleistein, Cohen, andStockwell 2000; Weglein et al. 2003).

Consider three 1D synthetic traces in the context of atime-lapse framework (Fig. 8 b,e,h). Suppose we surveyeda baseline medium consisting in a single reflector, produc-ing a trace with a single primary (Fig. 8 a,b). Linearizedinverse scattering (trace integration), if we grant ourselvesfull bandwidth data, generates a reasonable approximate re-construction of the interface (Fig. 8 c). Suppose further thatafter a certain interval of time, during which the interfacehas migrated upward, we repeat this experiment. Again lin-earized inverse scattering transforms the spike data into astep interface, and generates a reasonable approximation



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 8 A 1D example of linearized inverse scattering applied to time-lapse data. (a) A single interface at the time of the baseline survey; (b)seismic data due to the baseline model consisting in a single primary; (c) the interface reconstructed through linearized inverse scattering (solid)versus the exact interface (dashed). (d) The same interface having migrated upward by the time of the monitoring survey; (e) the associatedprimary, which arrives at an earlier time; (f) the interface reconstructed using linearized inverse scattering (solid) versus the exact interface(dashed). (g) A representation of the time-lapse change in the model; (h) the time-lapse data calculated by differencing (e)–(b); (i) the linearizedreconstruction of the difference model (solid) through trace integration of (h), versus the exact representation (dashed).

of the medium, including its newly repositioned reflector(Fig. 8 d-f). Finally, let us employ the time-lapse/scatteringidea, in which the baseline model and the monitoring model,each with a single interface, become, respectively, the refer-ence and perturbed media. The difference data, produced bysubtracting the baseline field from the monitoring field, con-sist in a trace with one ‘up’ spike and one ‘down’ spike (Fig.8 h). Applying the same linearized inverse scattering/traceintegration algorithm as before to the difference data tracegenerates a ‘bump’ model (Fig. 8 i). This, when transformedto wavespeed units, faithfully approximates the differencebetween the baseline and monitoring media–a positive re-sult. Linearized inverse scattering methods appear to havea strong potential for direct imaging of time-lapse difference

data, generating maps of the structural changes occurring in areservoir.

But it is a difficult result to explain. The bump is builtfrom the integration of the two primaries in the differencetrace (Fig. 8 h). Both primaries are needed to do the building:in integrating from left to right, the left-hand primary ‘turnson’ the bump, and the right-hand primary ‘turns it back off’.The second primary comes from the reflection in the referencemedium, i.e., the baseline model. But, the inverse scatteringalgorithm we used–the trace integration–was derived underthe assumption of a homogeneous reference medium, whichcauses no reflections. How could we be getting anything likethe right answer, when half of the contributing events do notexist, from the point of view of the theory?



We are left with the sense that, although existing imag-ing methods derived from inverse scattering may be success-fully applied to time-lapse difference data, the theory withwhich those algorithms are derived sheds little light on whythey work. Let us look to a consistently posed time-lapseversion of the theory not merely as a source of algorithms,but as a way of providing intelligible explanations of theirbehaviour.

Initial analysis

We will end this paper by examining both a linearized (first-order) model of time-lapse difference data, and its inversion,configured to produce images of time-lapse structural changes.Before doing so, we will reconsider the 1D scalar environmentthat was used to analyze the immobile interface problem, herealtering it to conform to a mobile interface problem. In thissection, the initial analysis serves not to establish new time-lapse specific techniques, but rather to show that existingmethods, derived outside the time-lapse problem and basedon assumptions that contradict those of the time-lapse pertur-bation framework, are in fact correct, to leading order, andwhy.

Scattering from a depth-perturbed interface

Let us set up a scalar time-lapse perturbation problem thatmirrors the ‘motivating example’ in Fig. 8. To accomplish this,the reference (baseline) medium and associated wavefield G

are chosen to satisfy[d2

dz2+ ω2

c2BL(z)

]G(z, zs) = δ(z − zs),

1c2

BL(z)=

{c−2

1 , z > zBL

c−20 , z < zBL

, (34)

where c0 is the incidence medium scalar velocity and c1 that ofthe target. The interface causing the single reflection is at depthzBL and the reflection coefficient associated with this interfaceis R1 = (c1 − c0)/(c1 + c0). The perturbed (monitoring) fieldP meanwhile satisfies[

d2

dz2+ ω2

c2M(z)

]P(z, zs) = δ(z − zs),

1c2

M(z)=

{c−2

1 , z > zM

c−20 , z < zM

. (35)

The incidence and target scalar velocities remain unchangedbut the location of the interface has moved to zM. The pertur-

bation must consequently reflect the migration of the bound-ary. We express it as

ac(z) = 1 − c2BL(z)

c2M(z)

= ac[H(z − zM) − H(z − zBL)], (36)

where ac = 1 − c2BL/c2

M is the size of the perturbation withinthe layer. Again in a 1D scalar environment with a singleinterface separating two homogeneous half-spaces, an exactGreen’s function G = P (0) is available and so with this ref-erence field and a form for the perturbation, terms for theexpansion of P may be directly calculated. In Appendix D wecarry out this expansion, in particular showing that if onlyfirst-order terms are retained P may be approximated as

P ≈ 1i2k0

+ R1ei2k0zBL

i2k0︸︷︷︸reference field˜P(0)

+ ac

4

[ei2k0zM

i2k0− ei2k0zBL

i2k0

]︸︷︷︸

1st order correction˜P(1)

, (37)

with zg = zs = 0. In equation (37) we have indicated con-tributions at zeroth and first order in the time-lapseperturbation.

On the appropriateness of non time-lapse imaging methods

The first of the four terms on the right of equation (37) ispart of the reference field, and corresponds to the direct wavebetween the (collocated) source and receiver. It is a physicallymeaningful part of both the reference and perturbed fields,and it is correctly conferred onto P directly by P (0). The sec-ond term on the right is the reflection from the interface zBL.This reflection occurs in the reference medium, and is also con-ferred onto P by P (0). Unlike the direct wave, however, thepresence of this second term is problematic, because there is nosuch event in the monitoring field P. If the full series derivedin Appendix D is to produce the right answer, this spuriousevent must be deleted through the activity of the higher orderterms.

Consider next the two rightmost terms. The first of theseis a reflection from the perturbed depth, zM, and is the lin-ear term in a series constructing the (correct) single primaryreflection in P. The second of these has the phase of the ref-erence reflection, and a negative sign–it is the linear termin a second, coupled, series, whose equally important ob-jective is to delete the reference reflection from the final,summed, result. So, when we take this series for P and sub-tract from it P (0) to obtain the scattered field, we are left withtwo terms of different phase, both linear in ac and with op-posite polarity. Assuming a homogeneous wave velocity of



c0, their phases correspond with the events in the differencedata (Fig. 8 h).

This provides a plausible explanation for the earlierquandary. A consistently posed linear time-lapse scatteringexpression correctly predicts the reflected events due to thereference-baseline medium, including their negative polar-ity.This happens because the series terms are extinguishing thereference events from the perturbed field. Further, within thelinear approximation, the phase of these events is associatedwith the homogeneous velocity of the medium in which thesources are embedded. In fact, the mobile interface time-lapsescattering problem is, linearly, indistinguishable from a two-interface non time-lapse problem with a homogeneous refer-ence. This provides ample reason to expect a trace-integrationprocedure to generate an approximation to the correct ‘bump’shaped difference model.

Indeed it is suggestive that ‘existing’ imaging algorithmsbased on linear inverse scattering are (fortuitously) directlyapplicable to time-lapse difference data. Error should onlyappear as the time-lapse changes grow in contrast.

A multidimensional model of time-lapse difference data

We proceed, building on the above conclusion, to apply alinear inverse scattering imaging algorithm to synthetic time-lapse difference data. Of all such algorithms in principle avail-able, we adopt one with the flexibility to, ultimately, regularizethe time-lapse imaging problem as discussed in the introduc-tion. A multidimensional scalar model relating time-lapse dif-ference data Dd to the difference model ac is, by our previousarguments,

Dd(r, rS, ω) ≈ S(ω)∫

dr′G(r, r′, ω)ω2

c20(r′)

ac(r′)G(r′, rS, ω),

(38)

where S(ω) is the source wavelet signature and c20(r) is a

smooth background velocity in agreement with that in whichthe sources and receivers are embedded. Imaging can beviewed as the exact or approximate inversion of this rela-tionship. What follows is a brief outline and simplified im-plementation of the shot-profile algorithm due to Kaplan,Naghizadeh, and Sacchi (2010a); Kaplan, Routh, and Sacchi(2010b).

Least-squares shot-profile formulation

The reference medium c20(r) is divided into layers of constant

thickness �z to allow for vertical variations in velocity. If de-

sired, lateral variations are incorporated through a split-stepmethod; we have not included either of these as yet in our 4Dapplication, rather, we consider a homogeneous backgroundvelocity c0. The forward operator in equation (38) as it ap-pears in this framework expresses the data Dd = Dd(kg, ω; xs)as

Dd = ω2

c20

�z∑

l

vr (l)(kg, ω)F[F∗vs(l)(kg, ω; xs)]ac(xg, zl ; xs) (39)

and the adjoint constructs a†c = a†

c (xg, zl ; xs) via

a†c = �ω

∑j

ω2j

c20

[F∗v∗s(l)(kg, ω j ; xs)]F

∗v∗r (l)(kg, ω j ; xs) (40)

where F and F∗ are the Fourier transform and inverse Fouriertransform respectively, vs(·) and vr (·) are defined in the spatialFourier domain by

vs(1)(kg, ω j ; xs) = up1(kg, ω j )g(kg, xs, ω j ),

vs(l)(kg, ω j ; xs) = �up(kg, ω j )vs(l−1)(kg, ω j ; xs),

vr (1)(kg, ω j ; xs) = up1(kg, ω j )Dd(kg, xs, ω j ), (41)

vr (l)(kg, ω j ; xs) = �up(kg, ω j )vr (l−1)(kg, ω j ; xs)

and where up1= exp(ikgz

(z1 − z0))/(i4kgz), �up =

exp(ikgz�z), g = 2π S(ω) exp(−ikg · xg), kgz

= (ω2/c20 −

kg · kg)1/2 and S is the source wavelet. Equations (39)–(41)define a forward modelling operator L and its adjointLH. If one shot record of difference data Dd(kg, xs, ω)is contained in the vector dobs and the time-lapse per-turbations ac are contained in the model vector ac, theleast-squares problem may be cast as the minimization of�(ac) where

�(ac) = ||Wd(dobs − Lac)||22 + μ||Wmm||22 (42)

and where Wd and Wm are prescribed data weights and modelweights respectively; or, equivalently, as the solution of thesystem of normal equations

[LHWHd WdL + μWH

mWm]ac = LHWHd Wddobs . (43)

Synthetic example: time-lapse variations in the scalar velocityof an anticline

Figure 9 illustrates some sample shot gathers to be used asinput for a 2D synthetic example of least-squares shot-profileimaging of a time-lapse difference structure, from a baselinesurvey (Fig. 9 a), a monitoring survey (Fig. 9 b) and the differ-ence of the two (Fig. 9 c).The synthetic data were generatedusing a scalar finite difference algorithm, with a zero-phase



(a)

0.5

1.0

1.5

Tim

e (s

)

1 5 9 13 17 21 25 29Shot number

(b)

0.5

1.0

1.5

Tim

e (s

)

1 5 9 13 17 21 25 29

(c)

0.5

1.0

1.5

Tim

e (s

)

1 5 9 13 17 21 25 29

Figure 9 Synthetic shot gathers corresponding to (a) baseline data, (b) monitoring data, and (c) difference data, corresponding to the baselineand monitoring models in Fig. 10(a,b.).

Ricker wavelet source (central frequency 15 Hz). If Fig. 9(c) ischosen to be the scattered field projected onto a measurementsurface, then to within a linear approximation the reconstruc-tion of the associated perturbation is a reconstruction of the

difference structure accruing in the Earth volume betweensurveys.

Figures 10(a,b) illustrates the Earth volumes. A produc-tion, injection, or spontaneous/natural process is modelled as



(a)

250

500

750

Dep

th (

m)

250 750 1250Distance (m) (b)

250

500

750

250 750 1250Distance (m)

1500 2000 2500 3000Velocity (m/s)

(c)

250

500

750

250 750 1250Distance (m)

Figure 10 Synthetic time-lapse imaging problem: (a) baseline medium, (b) monitoring medium, after a notional production process has alteredthe P-wave velocity at the vertex of the anticline; (c) least-squares shot profile imaging of the difference data in Figure 9(c.).

(a)

550

650

750

Dep

th (

m)

600 800 1000Distance (m)

1500 2000 2500Velocity (m/s)

(b)

550

650

750

600 800 1000Distance (m) (c)

Wav

elet

Figure 11 Detail imaging results. (a) Close-up of time-lapse target; (b) variable-area close-up of imaging result; (c) illustration of the zero-phaseRicker wavelet used as a source.

occurring at the vertex of the deeper of the two anticlines.Figure 10(c) is the result of applying the procedure embodiedin equations (39)–(42), which confirms the ability of the ap-proach to image and characterize the region of 4D change.InFig. 11(a,b) we illustrate the regions of change in more de-tail, both in the synthetic model (a) and the derived image(b). In Fig. 11(c) we illustrate the source wavelet used inthe simulation, a zero-phase Ricker. We emphasize that lit-tle of the potential for regularization contained in this ap-proach has been exploited as yet, nor have synthetic instancesof repeatability problems yet been included in the differencedata.

CONCLUSIONS

A self-consistent, full-wavefield, perturbation description ofthe time-lapse seismic reflection problem is likely impos-sible to achieve in general, since a full knowledge of thebaseline Earth volume and its associated Green’s functionare required. However, difference-AVO scenarios and lin-earized difference data imaging scenarios comprise two spe-cial cases within which practically useful methods can beexpected to emerge. It is appealing to have both typesof time-lapse method emerge from a common theoreticalframework.



In the AVO setting difference-amplitudes are analysed. Acomplete description of the difference amplitude involves bothbaseline quantities (in the form of interface contrast values orreflection coefficients) and time-lapse quantities. The latterbegin to contribute at first order and the former at secondorder but given large contrasts, both are necessary for even thelow-angle behaviour of �R to be adequately captured. This istrue for one parameter scalar, two parameter acoustic and twoparameter anacoustic problems and is likely true in general.

In the linear/low-contrast scenario we conclude that alinearized inverse scattering framework of the kind commonlyused in non time-lapse imaging applications is appropriate fordifference data. Numerical testing in a 2D scalar environmentconfirms this. A least-squares migration algorithm formulatedbased on the time-lapse perturbation model provides, inprinciple, a practical means to choose optimum images in thepresence of the missing data common to time-lapse data sets.

Ongoing research includes an extension of the ampli-tude/AVO difference methods to elastic and/or anelastic casesand investigation of data differencing techniques that, in com-bination with the least-squares imaging formulation, will formdifference data input of adequate fidelity for modelling andinversion methods of this kind.

ACKNOWLEDG E ME N T S

This research was supported by the Consortium for Researchin Elastic Wave Exploration Seismology (CREWES) and anNSERC Discovery grant.

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Figure A1 Illustration of the action of the 1-interface Green’s functions (equations (A1)–(A4)) within the zeroth-, first- and second-order termsof the time-lapse wave calculation.

APPENDIX A

So lu t i on s Fo r an Amp l i t ude -P e r tu rbed In t e r f a c e

In this Appendix we will discuss some of the details of the 1Dscalar problem for the immobile interface with changing targetvelocity. The ingredients for the expansion are the Green’sfunctions for wave propagation in the reference medium, andthe perturbation describing the change from the reference tothe perturbed medium. The perturbation is given in equation(3). The Green’s function for a 1D scalar medium with a singleinterface is available exactly and in closed form, though it hasseveral possible forms depending on the locations of the sourceand observation points zs and zg relative to the interface depthz1 (see Fig. A1). If zg, zs < z1, the Green’s function, which wewill label 00, is

G00(zg, zs) = eik0|zg−zs |

i2k0+ RBL

eik0(z1−zg )eik0(z1−zs )

i2k0, (A1)

where k0 = ω/c0 and RBL = (cBL − c0)/(cBL + c0). If zg, zs > z1

the Green’s function is labelled 11 instead and has the form

G11(zg, zs) = eikBL|zg−zs |

i2kBL− RBL

eikBL(zg−z1)eikBL(zs−z1)

i2kBL, (A2)

where kBL = ω/cBL. If zg > z1, zs < z1 we have

G01(zg, zs) = TDeik0(z1−zs ) eikBL(zg−z1)

i2k0, (A3)

where TD = 2cBL/(c0 + cBL), and if zg < z1, zs > z1 we have

G10(zg, zs) = TUeikBL(zs−z1) eik0(z1−zg )

i2kBL, (A4)

where TU = 2c0/(c0 + cBL). With these ingredients in place, wecalculate the first three terms in the Born series expansion

P(zg, zs) = P (0)(zg, zs) + P (1)(zg, zs) + P (2)(zg, zs) + · · · (A5)

Setting zg = zs = 0, we have P (0) = G00(0, 0) at zeroth orderin ac,

P (1) = ack2BL

∫ ∞

z1

dz′G10(0, z′)G01(z′, 0), (A6)

at first order and

P (2) = a2c k4

BL

∫ ∞

z1

dz′G10(0, z′)∫ ∞

z1

dz′′G11(z′, z′′)G01(z′′, 0),

(A7)

at second order, etc. Summing these, we have for the moni-toring field P

P =[1 + ei2k0z1

(RBL + ac

4TDTU

+ a2c

16TDTU(2 − RBL) + · · ·

)]1

i2k0. (A8)

The origins of each element of this series in terms of the var-ious parts of G are illustrated in Fig. A1. Finally, expressingthe baseline transmission coefficients in terms of the base-line reflection coefficient, TDTU = 1 − R2

BL and forming thetime-lapse difference field �P = P − P (0), we obtain the finalresult:

�P =[

14

ac

(1 − R2

BL

)

+18

a2c

(1 − 1

2RBL − R2

BL + 12

R3BL

)+ · · ·

]ei2k0z1

i2k0. (A9)



APPENDIX B

Expans ion o f th e D i f f e r enc ed Acous t i cReflec t i on Coe ffic i en t

In this Appendix we derive the expansion in equation (20). Webegin with the two parameter acoustic reflection coefficient atthe time of the monitoring survey:

RM(θ ) = 1 − �M(θ )1 + �M(θ )

, (B1)

where

�M(θ ) =(

ρ0

ρM

) (κ0

κM

ρM

ρ0

)1/2

(1 − sin2 θ )−1/2

(1 − κM

κ0

ρ0

ρMsin2 θ

)1/2

, (B2)

or, for angles well below critical,

�M(θ ) ≈(

ρ0

ρM

) (κ0

κM

ρM

ρ0

)1/2

[1 + 1

2sin2 θ

(1 − κM

κ0

ρ0

ρM

)]. (B3)

Multiplying and dividing several times by the baseline modu-lus (κBL) and density (ρBL),

�M(θ ) ≈(

ρ0

ρBL

ρBL

ρM

) (κ0

κBL

κBL

κM

ρ0

ρBL

ρBL

ρM

)1/2

[1 + 1

2sin2 θ

(1 − κBL

κ0

κM

κBL

ρBL

ρ0

ρM

ρBL

)](B4)

and substituting the four perturbations in equation (19), weproduce a form analogous to equation (11), which, therefore,may be expanded in the same way, such that the differencereflection coefficient

�R(θ ) = RM(θ ) − RBL(θ ) (B5)

has the form

�R(θ ) = Aκ (θ )aκ + Aρ(θ )aρ + Aκκ (θ )a2κ

+Aκρ(θ )aκaρ + Aρρ(θ )a2ρ + · · · , (B6)

where, neglecting terms at third order in bκ and/or bρ andhigher,

Aκ (θ ) ≈ 14

{1 + [

1 + (bκ − bρ)]

sin2 θ}

+ 164

[(b2

ρ − 18bκbρ + 13b2κ

)sin2 θ − (bκ + bρ)2] ,

Aρ(θ ) ≈ 14

{1 − [

1 + (bκ − bρ)]

sin2 θ}

+ 164

[(3b2

ρ + 18bκbρ − 17b2κ

)sin2 θ − (bκ + bρ)2] (B7)

and

Aκκ (θ ) ≈ 18

(1 + 2 sin2 θ

) − 164

(bκ + bρ)

+ 164

(13bκ − 17bρ) sin2 θ

Aκρ(θ ) ≈ −14

sin2 θ − 132

(bρ + bκ ) + 932

(bρ − bκ ) sin2 θ

Aρρ(θ ) ≈ 18

− 164

(bκ + bρ) + 164

(bκ + 3bρ) sin2 θ. (B8)

As a practical matter �R can be obtained by expanding themonitoring reflection coefficient alone, and then setting allterms in bκ and/or bρ alone to zero. The more complete deriva-tion, as described in the initial analysis section, is to expandRBL(θ ) in those two perturbations, and then subtract the twoseries, but the final effect is the same and the former is muchless laborious.

APPENDIX C

Expans ion o f th e D i f f e r enc ed Anacous t i cReflec t i on Coe ffic i en t

In this Appendix we derive the expansion in equation (32).The approach is identical to that of Appendix B. If a nearly-constant Q model is chosen to describe the target medium,following Aki and Richards (2002), the target velocity, say,cM, is replaced by

cM

[1 − F (ω)

QM

], (C1)

where F (ω) = i/2 − (1/π ) log(ω/ω0). Thus the reflection coef-ficient becomes complex and frequency dependent (e.g., Ode-beatu et al. 2006; Innanen 2011):

RM(θ, ω) = 1 − �M(θ, ω)1 + �M(θ, ω)

, (C2)

where, assuming a small angle of incidence,

�M(θ, ω) ≈(

c0

cM

)(1 + F (ω)

QM

){

1 + 12

sin2 θ

[1 − c2

M

c20

(1 − 2

F (ω)QM

)]}. (C3)

In order to reproduce the time-lapse expansion in equation(32), we make use of the perturbations defined in equation(33), and the rearrangement

�M ≈(

c0

cBL

cBL

cM

) (1 + F (ω)

QBL

QBL

QM

){

1 + 12

sin2 θ

[1 − c2

M

c20

(1 − 2

F (ω)QBL

QBL

QM

)]}. (C4)



Figure C1 Zeroth-, first- and second-order scattering interactions in the mobile-interface scattering problem. The reference (i.e., baseline)medium includes a reflector with strength R1 at depth zBL. Each path, beginning at a grey dot and ending at a black dot, contributes one termto the increasingly complex set of terms in equations (D5)–(D6). The number of white dots in each diagram indicates the order.

By expanding RM alone in the four anacoustic perturbationsand setting all terms in bc and/or bQ to zero, we producethe same results as if we had expanded both RM(θ, ω) andRBL(θ, ω) and then taken their difference:

�R(θ, ω) = 14

(1 + sin2 θ

)ac +

[14

bc − 12

F (ω)bQ

]sin2 θ ac

+ 12

(1 + sin2 θ )F (ω)bQaQ

+ 18

(1 + 2 sin2 θ )a2c + · · · . (C5)

APPENDIX D

So lu t i on s fo r a Dep th -Pe r tu rbed In t e r f a c e

In this Appendix we calculate scattering terms for a time-lapse problem involving a reflecting interface that moves froman initial (baseline) depth zBL to a final (monitoring) depthzM (Fig. C1). Let us assume for simplicity that zM < zBL. Inthis circumstance, the depth support of the perturbation is‘above’ the interface that causes the baseline reflection R1.Hence, though the same Green’s function used in Appendix Ais applicable here, only one ‘version’ of it is needed, the onein which the source and receiver are at depths less than zBL:

G(zg, zs) = eik0|z−zs |

i2k0+ R1

ei2k0zBL

i2k0e−ik0(z+zs ), (D1)

where k0 = ω/c0 and R1 = (c1 − c0)/(c1 + c0). Using the per-turbation ac(z) defined in equation (36) and setting zg = zs =0, which we may do without loss of generality, we expand P

in a Born series, obtaining

P = P (0) + P (1) + P (2) + · · · , (D2)

where P (0) = G(0, 0) and

P (1) = acω2

c20

∫ zBL

zM

dz′G(0, z′)G(z′, 0) (D3)

and

P (2) = a2cω4

c40

∫ zBL

zM

dz′G(0, z′)∫ zBL

zM

dz′′G(z′, z′′)G(z′′, 0) (D4)

etc. Using equation (D1) to substitute for the Green’s func-tions, we find

P (1) = ac

4ei2k0zM

i2k0− ac

4ei2k0zBL

i2k0(1 − R2

1) (D5)

− ac

4ei2k0zBL

i2k0[i4k0(zBL − zM)R1 + R2

1ei2k0(zBL−zM)]

and

P (2) = a2c

8ei2k0zF

i2k0− a2

c

8ei2k0zBL

i2k0

[(1 + R1 − R2

1 − 12

R31

)+ [2R2

1(i2k0)(zBL − zM) + (1 − R1 − R21)R1]ei2k0(zBL−zM)

+R1(i2k0)2(zBL − zM)2 + (i2k0)(zBL − zM)(1 − R1 − R21)

+12

R31ei4k0(zBL−zM)

](D6)

etc. Although the contributions to P evidently become quitecomplicated as the order increases, the origins of each partof each order can be straightforwardly traced to paths fromone scattering interaction to another, as elucidated in Fig.C1. In this paper we are concerned only with the first-ordercontributions, i.e., terms in equation (D5) with R1 set to zero.(With, that is, the order of a term aN

c × RM1 being defined

as N + M.) The meaning and usefulness of the higher orderterms will be the subject of future communications.



Table D1 TABLE OF SYMBOLS.

Symbol Meaning

δ(z), H(z) Delta function, Heaviside functionω Angular frequency (rad/s)c0 P-wave velocity of the incidence mediumcBL, QBL P-wave velocity, quality factor of the baseline target mediumcM, QM P-wave velocity, quality factor of the monitoring target mediumκ0, ρ0 Bulk modulus, density of the incidence mediumκBL, ρBL Bulk modulus, density of the baseline target mediumκM, ρM Bulk modulus, density of the monitoring target mediumbc Baseline perturbation in P-wave velocity, scalar 1D caseac, aκ , aρ , aQ Time-lapse perturbations in velocity, bulk modulus, density and Qacn

nth order term in the 1D inverse series for ac

zg , zs Receiver, source depthsθ Plane wave angle of incidenceRBL(θ ) Baseline reflection coefficient, immobile-interface caseRM(θ ) Monitoring reflection coefficient, immobile-interface caseR1 Baseline reflection coefficient, scalar, mobile-interface case�R(θ ) Difference reflection coefficient�R1, �R2 Vectors containing input difference reflection coefficientsk0 wavenumber k0 = ω/c0

kBL wavenumber kBL = ω/cBL in the baseline target mediumP(zg, zs ) 1D monitoring wavefield measured at zg due to source at zs

G(zg, zs ) 1D baseline (reference) Green’s functionG00(zg, zs ) Green’s function with zg and zs both in the incidence (0) mediumG11(zg, zs ) Green’s function with zg and zs both in the target (1) mediumG10(zg, zs ) Green’s function with zg in medium 1 and zs in medium 0G01(zg, zs ) Green’s function with zs in medium 1 and zg in medium 0TU , TD Up-, downgoing transmission coefficients (scalar, mobile-interface case)P (n)(zg, zs ) nth-order term in series expansion of P(zg, zs )z1 Interface depth, scalar immobile interface casezBL Baseline interface depth (scalar, mobile-interface case)zM Monitoring interface depth (scalar, mobile-interface case)Aκ (θ ), Aρ (θ ) Coefficients for �R(θ ) series terms, linearAκκ (θ ), Aκρ (θ ), Aρρ (θ ) Coefficients for �R(θ ) series terms, second orderac Model vector containing time-lapse perturbation valuesF, F∗ Forward, inverse Fourier transform operatorsS Source waveletWd, Wm Data and model weighting matricesμ Least-squares data misfit versus model norm trade-off parameter


Perturbation methods for two special cases of the time-lapse seismic inverse problem

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Transcript of Perturbation methods for two special cases of the time-lapse seismic inverse problem