generalized representation theorems for acoustic wavefields in

Geophys. J. Int. (2009) 179, 319–332 doi: 10.1111/j.1365-246X.2009.04266.x

GJI

Sei

smol

ogy

Generalized representation theorems for acoustic wavefields inperturbed media

Huub Douma∗Department of Geosciences, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected]

Accepted 2009 May 21. Received 2009 May 21; in original form 2008 May 18

S U M M A R YIn this paper, generalized representation theorems for acoustic wavefields in perturbed mediaare derived from convolution- and correlation-type reciprocity theorems. These theoremsallow calculation of the total wavefield in perturbed media from boundary integrals involvingknowledge of the impulse responses due to monopole and dipole sources at the boundary only.The presented derivation makes explicit the close connection between the convolution-typereciprocity theorem and the Lippmann–Schwinger integral equation for the scattered field.The derivation of the correlation-type representation theorem leads to an expression for thetotal wavefield analogous to the Lippmann–Schwinger equation, but expressed in terms ofthe time-advanced Green’s function instead. The generalized correlation-type representationtheorem provides an alternative method to obtain virtual-source time-lapse seismic wavefieldscompared to the current method of differencing base and monitor virtual-source wavefields,while the convolution-type theorem has potential applications in non-linear inversions andtime-lapse seismic feasibility studies for reservoir monitoring.

Key words: Interferometry; Theoretical seismology; Wave scattering and diffraction; Wavepropagation.

I N T RO D U C T I O N

A reciprocity theorem is a relation between two independent physical states in a common domain. In the context of seismic wave propagation,such reciprocity theorems have recently received a lot of attention in relation to seismic interferometry. Seismic interferometry refers to themethod of cross-correlating wavefields recorded at two receiver locations to obtain the Green’s function between both locations. Originallythis principle was shown by Claerbout (1968) for the special case of horizontally layered or 1-D media, who then went on and conjecturedthat this principle holds also for the 3-D case (Rickett & Claerbout 1999). Claerbout’s conjecture was later proven by Wapenaar (2004) usinga reciprocity theorem while Snieder (2004) provided a more intuitive derivation based on stationary phase. Wapenaar’s proof was recentlyextended to include diffusion, flow and general wave phenomena (Wapenaar et al. 2006; Snieder et al. 2007). There are many applicationsof interferometry. Without being complete, I mention crustal seismology (Campillo & Paul 2003; Sabra et al. 2005; Shapiro et al. 2005;Stehly et al. 2006), exploration seismology (Schuster 2001; Schuster et al. 2004; Bakulin & Calvert 2006; Xiao et al. 2006), helioseismology(Duvall et al. 1993; Rickett & Claerbout 2000), ultrasound (Weaver & Lobkis 2001; Borcea et al. 2005), civil engineering (Snieder & Safak2006), volcano monitoring (Sens-Schonfelder & Wegler 2006) and finally modelling of wave propagation (van Manen et al. 2005, 2006).

In most applications of seismic interferometry the reciprocity theorem underlying the method assumes that the medium parameters inboth states are equal. The most general form of a reciprocity theorem, however, allows the media in both states to be different. This allowsthe application of this theorem to the case of time-lapse monitoring, where the medium is assumed to change over time. That is, the mediumis changing in between a reference and monitor experiment, but is assumed stationary over the duration of each experiment individually.Wapenaar et al. (2001) and Dillen et al. (2002) used a reciprocity theorem of the convolution type for the one-way wave equation where themedium parameters in both states differ, and showed its use in determining the time-lapse changes in reflectivity from a boundary integral.Vasconcelos (2007) derives a representation theorem for the scattered field in perturbed media closely related to the representation theorems

∗Now at: ION Geophysical, GXT Imaging Solutions, 225 E. 16th Ave., Suite 1200, Denver, CO 80203, USA.

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320 H. Douma

presented here (but using an alternative derivation) and Vasconcelos et al. (2008) show the use of this theorem to image internal multiples inVSP data.

Here, I analyse the general form of the acoustic reciprocity theorem for the full, as opposed to the one-way, wave equation for allcases of possible source locations, that is, both inside or both outside the scattering (or perturbed) volume, or one inside and one outside. Ido this for both convolution-type and correlation-type reciprocity theorems and show that this leads to generalized acoustic representationtheorems for wavefields in perturbed media, where the perturbation can be of arbitrary scattering strength. I discuss the relation betweenthe generalized correlation-type representation theorem and the virtual source method (Bakulin & Calvert 2006) and point out a potentialuse of the generalized convolution-type representation theorems for efficient modelling of wave propagation in non-linear inversions or intime-lapse monitoring feasibility studies.

The outline of this paper is as follows. First, I analyse the convolution-type reciprocity theorem for all possible combinations of sourcepositions and derive for each case the resulting boundary-integral representations for the pressure Green’s function in a locally perturbedmedium. This derivation makes explicit the connection between the convolution-type reciprocity theorem and the Lippmann–Schwingerequation for the total wavefield. Subsequently, I provide an alternative derivation of the same result to make the connection with the existingliterature on this topic. This analysis leads to a generalized acoustic representation theorem of the convolution type for the total wavefield.In the process of analysing the convolution-type reciprocity theorem for the case when the receiver is inside the scattering volume, I find ageneralization of the Ewald–Oseen extinction theorem [see, for example, Born & Wolf (1970), pp. 101–102]. I then proceed to derive theequivalent generalized correlation-type representation theorem and present an equation for the total wavefield analogous to the Lippmann–Schwinger equation, but in terms of the time-advanced Green’s function instead of the time-retarded Green’s function. Finally, I illustratenumerically the validity of the generalized convolution-type representation theorem for the special case of 1-D wave propagation and mentionthe connection between the generalized correlation-type representation theorem and the virtual source method, as well as the potentialapplication of the generalized convolution-type representation theorems in non-linear inversions and wavefield modelling for time-lapsemonitoring feasibility studies.

A C O U S T I C C O N V O LU T I O N - T Y P E R E C I P RO C I T Y T H E O R E M

The governing equations for waves in non-flowing acoustic media are the linearized mass continuity equation, the linearized momentumequation, and the linearized equation of state, given by, respectively,

∂tρ1 + ∇ · (ρ0v1) = ρ0q, (1)

ρ0∂tv1 + ∇ p1 = 0, (2)

∂tρ1 + v1 · ∇ρ0 = 1

c2∂t p1, (3)

where ρ is the density in kg m−3, v is the particle velocity in m s−1, p is the pressure in kg m−1 s−2, q is a volume injection rate density sourcein s−1 and c is the propagation velocity in m s−1. These equations have been linearized; the subscript 0 indicates the equilibrium state, whilethe subscript 1 denotes the deviation from it. Note that in eqs (1)–(3) ρ0 = ρ0(x), since ∇ρ0 �= 0. Eqs (1)–(3) do not contain terms with p0

and v0 because the equilibrium state is assumed static and non-flowing, that is, v0 = 0. Combining eqs (1) and (3) by eliminating the term∂tρ1 results in

κ∂t p1 + ∇ · v1 = q, (4)

where κ = 1/(ρ0c2) denotes the compressibility. This is the stress–strain relation for an acoustic medium, where the volume injection ratedensity source q plays the role of the time-derivative of the stress–glut, that is, the glut rate. Taking the Fourier transform with respect to timeof eqs (2) and (4) I get the known system of coupled first-order partial differential equations given by

iωρv + ∇ p = 0, (5)

iωκ p + ∇ · v = q, (6)

where I dropped the subscripts 0 and 1 and where the hat denotes the Fourier transformation with respect to time; that is, p and v here referto deviations from the static equilibrium state whereas ρ and κ are the medium parameters from the equilibrium state. Attenuation losses areincluded when κ is complex-valued. Throughout this paper, I adopt the following Fourier transform convention

f (x, ω) =∫ ∞

−∞f (x, t)e−iωt dt , f (x, t) = 1

2π

∫ ∞

−∞f (x, ω)eiωt dω . (7)

Consider the interaction quantity (de Hoop 1988)

pAvB − vA pB, (8)

where the subscripts A and B denote two different states A and B of the wavefield. Throughout this paper I assume that the density andcompressibility of the medium in state A and B are different. Taking the divergence of the interaction quantity, using the chain-rule, and

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Generalized representation theorems for acoustic wavefields in perturbed media 321

subsequently using eqs (5) and (6) to eliminate the terms ∇ pA,B and ∇ · vA,B, respectively, it follows that

∇ · ( pAvB − vA pB) = pAqB − pBqA + iωvA · vB�ρ − iω pA pB�κ, (9)

where I defined

�ρ := ρB − ρA, �κ := κB − κA. (10)

Integrating both sides over an arbitrary volume D with boundary ∂D and outside pointing normal n, and using Gauss’ theorem, it followsthat∫

x∈∂Dn · ( pAvB − vA pB) dx =

∫x∈D

( pAqB − pBqA + iωvA · vB�ρ − iω pA pB�κ) dx. (11)

This is a reciprocity relation of the convolution type relating the wavefields in state A and state B. This theorem is almost the same as theglobal form of lord Rayleigh’s reciprocity theorem (Rayleigh 1945) given by (Fokkema & van den Berg 1993, their eq. 5.7, p. 97) except forthe missing dipole body-force sources f A and f B; these can be obtained by setting the right-hand side of eq. (2) equal to f , and thus theright-hand side of eq. (5) to f . I ignore these dipole sources because I derive representation theorems for the pressure only. Note that volumeD in eq. (11) is arbitrary.

In the rest of this study, I will need two important known results: the representation theorem for the acoustic field and the Lippmann–Schwinger integral equation for scattering in heterogeneous media. For the sake of being complete and to make sure that all quantities aretreated consistently throughout, I briefly digress from the main argument to rederive these fundamental equations. I start with the representationtheorem and derive it from the general reciprocity theorem given in eq. (11).

C O N V O LU T I O N - T Y P E R E P R E S E N TAT I O N T H E O R E M

Consider the special case where the medium parameters in both states are equal. In that case the perturbations �ρ and �κ are zero everywhere.Hence, the reciprocity theorem in eq. (11) reduces to

− 1

iω

∫x∈∂D

1

ρ(x)n · ( p1∇ p2 − p2∇ p1) dx =

∫x∈D

[q2 p1 − q1 p2] dx, (12)

where I replaced the subscripts A and B with 1 and 2, indicating that the fields p1 and p2 originate from sources at different locations x1 andx2, respectively, and where I used eq. (5) to find

v1,2 = − 1

iωρ∇ p1,2(x). (13)

Upon choosing

q1,2 = δ(x − x1,2) (14)

it follows by definition of the Green’s function that

p1,2 = G(x, x1,2). (15)

Throughout this paper, I use the shorthand notation G(x, x ′) for G(x, x ′, ω), where the first variable (x) denotes the receiver location and thesecond variable (x ′) the source location. Using eqs (14) and (15) in (12), I then find∫

x∈DG(x1, x)δ(x − x2)dx +

∫x∈∂D

n ·[

G(x, x1)∇G(x, x2) − G(x, x2)∇G(x, x1)

iωρ(x)

]dx =

{G(x2, x1), x1 ∈ D0, x1 /∈ D

.(16)

Instead of choosing the volume D as integration volume in eq. (12), we can also choose the complement of D, that is, D′ (see Fig. 1), as theintegration volume. Doing this while choosing the source locations x1,2 /∈ D′ it follows that∫

x∈∂D′n ·

[G(x1, x)∇G(x2, x) − G(x2, x)∇G(x1, x)

iωρ(x)

]dx = 0. (17)

Figure 1. Schematic representation of volume D and its complement D′, with the source locations x1,2 ∈ D.

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322 H. Douma

Recognizing that ∂D′ = ∂D ∪ ∂∞ (see Fig. 1), and that the contribution of the boundary ∂∞ at infinity is zero due to radiation boundaryconditions (or causality), it follows that∫

x∈∂Dn ·

[G(x1, x)∇G(x2, x) − G(x2, x)∇G(x1, x)

iωρ(x)

]dx = 0, (18)

where the integration is now only over the boundary ∂D. Using this in eq. (16) it follows that when x1,2 /∈ D′, and thus x1,2 ∈ D, we have

G(x1, x2) = G(x2, x1), (19)

which is the well-known source–receiver (sometimes referred to as physical or field) reciprocity. Using this reciprocity in the surface integralin eq. (16), we find the well-known representation theorem∫

x∈DG(x1, x)δ(x − x2)dx +

∫x∈∂D

n ·[

G(x1, x)∇G(x2, x) − G(x2, x)∇G(x1, x)

iωρ(x)

]dx =

{G(x2, x1), x1 ∈ D0, x1 /∈ D

, (20)

that includes source–receiver reciprocity if we realize that the surface integral vanishes when x1,2 ∈ D (as shown in eq. 18).Note that eq. (20) encompasses both cases where the receiver location x1 (sometimes referred to as the field point) is inside and outside

of the domain D. In the absence of sources (i.e. sources at x2) in D eq. (20) is known as the ‘integral theorem of Helmholtz and Kirchhoff’[see, for example, Baker & Copson (1950), pp. 24–25, or Born & Wolf (1970), p. 377]. This equation is the basis of Kirchhoff migration inseismic data processing (e.g Yilmaz 2001, pp. 1343–1346). For a more detailed and standard derivation of the representation theorem thatdoes not make explicit use of a reciprocity theorem, I refer the reader to the existing literature [for example, Aki & Richards (2002, pp. 38–42)or Cerveny (2001, pp. 89–95)]. When x1 /∈ D eq. (20) is known as the Ewald–Oseen extinction theorem (Oseen 1915; Ewald 1916; Born &Wolf 1970).

L I P P M A N N – S C H W I N G E R I N T E G R A L E Q UAT I O N

In order to treat the scattering of waves from inhomogeneities in a medium, we need to define a background (often referred to as reference)medium in which the inhomogeneities (or perturbations) are embedded. The wavefield is then naturally subdivided into a background wavefieldthat travels in the background medium and a scattered wavefield that originates from the scattering of the inhomogeneities superposed onthe background medium. To obtain a representation for the total wavefield in the medium including the scatterers, that is, the sum of thebackground and scattered wavefield, we choose state A in eq. (11) to be the wavefield of the background medium and state B the totalwavefield. Choosing

qA,B(x) = δ(x − x1,2), (21)

in eq. (11) with x1,2 ∈ D, it follows from the definition of the Green’s function that

pA,B = GA,B(x, x1,2), (22)

and from the equation of motion (5) that

vA,B = − 1

iωρA,B∇GA,B(x, x1,2). (23)

Using eqs (21)–(23) in eq. (11) it then follows that

GB(x1, x2) = GA(x1, x2) + 1

iω

∫x∈D

∇GA(x1, x) · ∇GB(x2, x)�ρ(x)

ρA(x)ρB(x)dx − iω

∫x∈D

GA(x1, x)�κ(x)GB(x2, x)dx

+ 1

iω

∫x∈∂D

n ·[

∇GB(x2, x)

ρB(x)GA(x1, x) − ∇GA(x1, x)

ρA(x)GB(x2, x)

]dx, (24)

where I again used source–receiver reciprocity and where �ρ and �κ denote the scattering inhomogeneities.Eq. (24) is called the Lippmann–Schwinger integral equation after its quantum-mechanical equivalent for scattering of particles from a

potential governed by the Schrodinger equation (see, for example, Schiff 1968, pp. 318–319). This equation clearly includes all non-linearinteractions between the perturbations (�ρ and �κ) and the pressure wavefield G, since G appears both on the left-hand side of the equationand in the integrands on the right-hand side. For a more detailed derivation of the Lippmann–Schwinger equation I refer the reader to (Morse& Feshbach 1953, section pp. 1072–1073; they do not refer to this equation as the Lippmann–Schwinger integral equation but instead referto it as the scattering integral equation).

C O N V O LU T I O N - T Y P E B O U N DA RY- I N T E G R A L R E P R E S E N TAT I O N S

Consider again the convolution-type reciprocity theorem in eq. (11). In this theorem, we are free to choose the volume injection rate densitysources qA,B. Choosing point sources in both states, qA,B, pA,B and vA,B are given by eqs (21)–(23), respectively. These point-sources act atlocations x1 and x2 in medium A and B, respectively. In the reciprocity theorem we are free to choose the integration volume and the locationof the sources. I choose this volume to be the volume V that encloses all areas where the media in state A and B differ, that is, all locations

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Figure 2. The volume D together with its boundary ∂D represent the domain of the unperturbed medium (state A). The enclosed local perturbation is nonzeroin V (with boundary ∂V) only. The normal vectors to both ∂D and ∂V are pointing outward.

where �ρ and �κ are non-zero (Fig. 2). This volume is embedded in volume D that represents the full spatial extent of the unperturbedmedium. This volume could be R

3 or a volume that extends to infinity but is partly bounded by a free surface. For the source locations, Iconsider the following four cases (Fig. 3)

(i) x1 /∈ V and x2 ∈ V (Fig. 3a)(ii) x1 ∈ V and x2 /∈ V (Fig. 3b)(iii) x1,2 /∈ V (Fig. 3c)(iv) x1,2 ∈ V (Fig. 3d).

Each of these cases will now be treated separately.Case (i): Using this choice of source locations together with eqs (21)–(23) in the reciprocity theorem (eq. 11), it follows that

1

iω

∫x∈∂V

n ·[

1

ρA(x)GB(x2, x)∇GA(x1, x) − 1

ρB(x)GA(x1, x)∇GB(x2, x)

]dx

= GA(x1, x2) + 1

iω

∫x∈V

∇GA(x1, x) · ∇GB(x2, x)�ρ(x)

ρA(x)ρB(x)dx

− iω∫

x∈VGA(x1, x)�κ(x)GB(x2, x)dx,

(25)

where I again used source–receiver reciprocity (cf. eq. 19). Comparison of this expression with the Lippmann–Schwinger integral eq. (24)now reveals that the right-hand side of this expression is almost equal to GB(x1, x2). Compared to the Lippmann–Schwinger integral equationthis expression does not have the remaining surface integral over ∂D while the volume integrals are taken over V instead of D. Recognizing,however, that we chose V to be the part of D where the perturbations �ρ and �κ are non-zero, the volume integrals over V and D are thesame. Therefore, we are left with the surface integral over ∂D. Assuming that the wavefield satisfies radiating boundary conditions on ∂D orthat ∂D is a free surface, the surface integral over ∂D in the Lippmann–Schwinger eq. (24) is zero. Note that this integral remains zero if partof the boundary ∂D is a free surface and the wavefield satisfies radiating boundary conditions at the remainder, as is the case in explorationseismics. With these boundary conditions for ∂D, it follows from the Lippmann–Schwinger eq. (24) that the right-hand side of eq. (25) indeedequals GB(x1, x2). Therefore, we have

GB(x1, x2) = 1

iω

∫x∈∂V

n ·[

1

ρA(x)GB(x2, x)∇GA(x1, x) − 1


]dx. (26)

This boundary-integral representation reveals that the Green’s function of the perturbed medium between any point x2 interior to theperturbed area and any point x1 outside of it, can be found by knowing the impulse responses and their derivatives for sources at the boundary∂V only. Note that the derivatives are with respect to the source location and that they thus represent the responses due to dipole sources.

Case (ii): When using x1 ∈ V and x2 /∈ V in the reciprocity theorem of eq. (11), the term GA(x1, x2) on the right-hand side ofeq. (25) is replaced with −GB(x1, x2). By adding and subtracting GA(x1, x2) on the right-hand side of the resulting expression and using theLippmann–Schwinger equation (24) with the same boundary conditions on ∂D as in case (i), it follows that

GA(x1, x2) = − 1

iω

∫x∈∂V

n ·[

1

ρA(x)GB(x2, x)∇GA(x1, x) − 1


]dx . (27)

In this case the boundary-integral equals the Green’s function between x1 and x2 for the unperturbed medium.Case (iii): When both x1,2 are outside of V , the term GA(x1, x2) on the right-hand side of eq. (25) disappears. Adding GA(x1, x2) on

both sides of the resulting expression and using the same reasoning that led to eq. (26) it follows that

GB(x1, x2) = GA(x1, x2) + 1

iω

∫x∈∂V

n ·[

1

ρA(x)GB(x2, x)∇GA(x1, x) − 1


]dx . (28)

Note that the only difference with case (i) is the appearance of the Green’s function in state A. Hence, this expression allows the Green’s functionof the perturbed medium between any two points exterior to the perturbed area to be calculated from a boundary-integral representation also,

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324 H. Douma

Figure 3. The point sources to be used in the reciprocity theorem (eq. 11) can be located in four possible ways: one source inside and one outside theperturbation area, that is, (a) x1 /∈ V and x2 ∈ V or (b) x1 ∈ V and x2 /∈ V , both sources outside the perturbation area, that is, (c) x1,2 /∈ V , or both sourcesinside the perturbation area, that is, (d) x1,2 ∈ V .

provided the Green’s function of the background medium between both points is known. This representation again contains impulse responsesand their derivatives from sources at the boundary ∂V only.

Case (iv): When both x1,2 are inside V , an extra term −GB(x1, x2) is introduced in the right-hand side of eq. (25) giving

1

iω

∫x∈∂V

n ·[

1

ρA(x)GB(x2, x)∇GA(x1, x) − 1


]dx

= GA(x1, x2) − GB(x1, x2) + 1

iω

∫x∈V

∇GA(x1, x) · ∇GB(x2, x)�ρ(x)

ρA(x)ρB(x)dx

− iω∫

x∈VGA(x1, x)�κ(x)GB(x2, x)dx. (29)

Again, applying the same reasoning and boundary conditions as used in cases (i)–(iii), it now follows that

1

iω

∫x∈∂V

n ·[

1

ρA(x)GB(x2, x)∇GA(x1, x) − 1


]dx = 0. (30)

Clearly, the boundary integral does not allow calculation of the Green’s function between two points interior to the perturbed area.The exact same procedure as outlined for the pressure Green’s function can be followed to get the Green’s function for the particle

velocity. In that case, we set the right-hand side of eq. (2) equal to a body-force density source f and subsequently use delta-function excitationsfor f while using q = 0. The same cases (i)–(iv) can then be treated.

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A LT E R NAT I V E D E R I VAT I O N

In the previous section, I showed the connection between the reciprocity theorem of eq. (11) and the Lippmann–Schwinger equation. Thisconnection is a consequence of the volume V I chose in the reciprocity theorem; because I chose V to be the volume that encompasses theperturbation, the perturbations �ρ and �κ are non-zero. As I mentioned earlier, the choice of the integration volume is, however, arbitrary.Hence an alternative (simpler) derivation is found by choosing the volume to be the complement of the perturbation, i.e. the complement V ′

in D of V . In that case �ρ and �κ are zero in the volume V ′ of integration, leading to a simpler reciprocity theorem. This is basically theapproach taken by Fokkema & van den Berg (1993). Of course the resulting boundary-integral representations I present should not dependon choosing either volume V or V ′ to begin with. I shortly recap this alternative approach to establish the connection with existing literature.

Let V ′ denote the complement of V in D (Fig. 2). Because �ρ and �κ are zero in V ′, the reciprocity theorem of eq. (11) simplifies to

−∫

x∈∂Vn · ( pAvB − pBvA) dx =

∫x∈V ′

(pAqB − pBqA) dx, (31)

where I used that ∂V ′ = ∂V ∪ ∂D and that the surface integral over ∂D is zero due to the radiating (or homogeneous) boundary conditions on∂D. The extra minus sign in front of the surface integral is due to the fact that the outward pointing normal n′ of volume V ′ is in the oppositedirection to the outward pointing normal n of volume V (Fig. 2).

Consider the following four cases that are equivalent to cases (i)–(iv) treated previously

(i) x1 ∈ V ′ and x2 /∈ V ′

(ii) x1 /∈ V ′ and x2 ∈ V ′

(iii) x1,2 ∈ V ′

(iv) x1,2 /∈ V ′.

Again I treat each of these cases separately while using the same choice of volume injection rate density sources (eq. 21) and the consequentforms of p and v (eqs 22 and 23, respectively).

Case (i): When only x1 is in volume V ′, the volume integral in eq. (31) contributes −GB(x1, x2) only. Therefore, using equations (22)and (23) for p and v, eq. (26) follows immediately from the reciprocity theorem for volume V ′ (eq. 31). The corresponding equation inFokkema & van den Berg (1993), is their eq. 8.50 on p. 153.

Case (ii): When only x2 is in V ′, the volume integral in eq. (31) contributes only GA. Hence eq. (27) follows directly from eq. (31).Fokkema & van den Berg (1993) do not treat this case.

Case (iii): When both x1 and x2 are in V ′, the volume integral in eq. (31) results in both GA and GB, hence giving eq. (28). Thecorresponding equation in Fokkema & van den Berg (1993), is their eq. 8.47 on p. 152.

Case (iv): When neither x1 or x2 are in V ′, the volume integral is obviously zero, giving us again eq. (30). Fokkema & van den Berg(1993) only treat this case when media A and B are the same [see their eq. (7.81) on p. 137].

G E N E R A L I Z E D A C O U S T I C C O N V O LU T I O N - T Y P E R E P R E S E N TAT I O N T H E O R E M

All cases (i)–(iv) can be combined into a single equation of the same form as the representation theorem in eq. (20). Doing this gives∫x∈V ′

GA(x1, x)δ(x − x2)dx +∫

x∈∂V ′n′ ·

[GA(x1, x)∇GB(x2, x)

iωρB(x)− GB(x2, x)∇GA(x1, x)

iωρA(x)

]dx =

{GB(x1, x2), x1 ∈ V ′

0, x1 /∈ V ′ , (32)

where the normal vector n′ points outward of V ′. Clearly all four cases treated previously are included in this single equation; that is, whenchoosing the respective positions of x1,2 for cases (i)–(iv), this equation reduces to eqs (26), (27), (28) and (30), respectively. Note that,contrary to the derivations in the previous sections, we have here not assumed radiating boundary conditions on ∂D as in the previoussections, as the bounding surface is here taken to be ∂V ′ = ∂V ∪ ∂D instead of ∂V only. Hence eq. (32) is the generalized convolution-typerepresentation theorem that is valid for arbitrary boundary conditions on ∂D.

Comparing eq. (32) and (20), it is clear that both equations have the same structure and that the only difference is in the Green’s functions.Therefore, eq. (32) is a generalization of the acoustic representation theorem for the total pressure field GB, where GA plays the role of theGreen’s function of the unperturbed (background) medium. In fact, in hindsight it follows that eq. (32) is just an application of eq. (20);just like there are no scattering contrasts in D in eq. (20), the integration volume V ′ in eq. (32) does not include any scattering contrasts.Therefore, reciprocity theorem (12) could have been invoked directly to derive representation theorem (32), just like representation theorem(20) followed from (12) directly.

The right-hand side of eq. (32) being zero when x1 /∈ V ′ does not mean that the field inside the scatterer is zero. In fact, when x1 /∈ V ′,that is, inside the scattering volume, the generalized representation theorem is the generalized form of the Ewald–Oseen extinction theorem(Oseen 1915; Ewald 1916; Born & Wolf 1970) for scattering of acoustic waves from a scattering object of arbitrary constitutive relations.For electro-magnetic waves this generalized form was shown by Sein (1970), Pattanayak & Wolf (1972) and de Goede & Mazur (1972),while the relationship between these independent derivations was later shown by Sein (1975). The classical interpretation of this theorem isthat the incident field at any point inside a scatterer is extinguished by part of the field originating from the dipole sources at the boundaryof the scatterer, the other part producing the field inside the scatterer [see, for example, Born & Wolf (1970), pp. 100–102]. That is, if

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326 H. Douma

a wave travelling with velocity c in the background medium is incident on a scatterer, this wave is extinguished inside the scatterer andreplaced with a wave travelling with velocity c/n due to the dipole sources at the scatterer boundary, where n is the refractive index of thescatterer. Fearn et al. (1996), however, point out existing conflicting interpretations of the theorem in the literature. Taking a microscopicview they show that the extinction is due to all dipole scatterers constituting the scattering volume instead of dipole sources at the boundaryonly, indicating that it is physically not justified to conclude that atoms or molecules at the boundary play a special role. In the field ofoptics the equivalents of the generalized representation theorem (eq. 32) for the electric and magnetic fields were derived using Green’stheorem by Pattanayak & Wolf (1972) and later again by Sanchez-Gil & Nieto-Vesperinas (1991). This theorem has found an application, forexample, in the study of light scattering from rough surfaces using numerical modelling (Maradudin et al. 1989; Sanchez-Gil & Garcıa-Ramos1998).

G E N E R A L I Z E D A C O U S T I C C O R R E L AT I O N - T Y P E R E P R E S E N TAT I O N T H E O R E M

Up until this point, I have made use of the convolution-type reciprocity theorem of eq. (11) only. This led to the generalized convolution-typerepresentation theorem in eq. (32). Alternatively, we can use a correlation-type reciprocity theorem as a starting point. Going through thesame motions as for the convolution-based derivations I then find a generalized correlation-type representation theorem. Considering that thederivation is essentially the same as for the convolution-based case, I will only summarize the main results and emphasize the differenceswith the convolution-based approach.

To find the correlation-type equivalent to the convolution-type reciprocity theorem of eq. (11), consider the interaction quantity

p∗AvB + v

∗A pB, (33)

where the star indicates complex conjugation. Going through the same steps as for the derivation that led to eq. (11), I find the correlation-typereciprocity theorem to be∫

x∈∂Dn · (

p∗AvB + v

∗A pB

)dx =

∫x∈D

(p∗

AqB + pBq∗A − iωv

∗A · vB�ρ − iω p∗

A pB

[κB − κ∗

A

])dx, (34)

where I used that the density is real. Using that κB − κ∗A = �κ + 2iImκA, I can rewrite this as∫

x∈∂Dn · (

p∗AvB + v

∗A pB

)dx =

∫x∈D

(p∗

AqB + pBq∗A − iωv

∗A · vB�ρ − iω p∗

A pB�κ)

dx + 2ω

∫x∈D

p∗A pBImκAdx. (35)

Apart from the fact that this equation contains complex conjugates of pA, vA and qA, the main difference between this equation and itsconvolution-type counterpart (cf. eq. 11) is the last term on the right-hand side. Only if medium A is non-attenuating this term is zero.When both media A and B are the same the correlation-type reciprocity theorem of eq. (35) reduces to eq. (4) of Snieder (2007). It is worthmentioning that the convolution-type reciprocity theorem is unaffected by the presence or absence of attenuation (Slob & Wapenaar 2007).The generalized convolution-type representation theorem of eq. (32) is therefore the same for both attenuating and non-attenuating acousticmedia.

Just as for the convolution-based approach, there are two possible choices for the volume, that is, V and V ′, to use in the reciprocitytheorem, and the final result should be independent of this choice. Choosing V ′ for the volume, using eqs (21)–(23) for qA,B, pA,B and vA,B,respectively, considering all four cases for the choices of x1,2, and gathering the results in one equation, it follows that

∫x∈V ′

G∗A(x1, x)δ(x − x2) dx +

∫x∈∂V ′

n′ ·[

G∗A(x1, x)∇GB(x2, x)

iωρB(x)− GB(x2, x)∇G∗

A(x1, x)

iωρA(x)

]dx

+ 2ω

∫x∈V ′

G∗A(x1, x)GB(x2, x)ImκA(x) dx =

{−GB(x1, x2), x1 ∈ V ′

0, x1 /∈ V ′ . (36)

Apart from the complex conjugates and the extra minus sign, the main difference with the generalized convolution-type representationtheorem is again the last term on the left-hand side that accounts for attenuation in the background medium A.

Going through all the same steps again but instead choosing volume V in the reciprocity theorem, I find

∫x∈V

G∗A(x1, x)δ(x − x2) dx +

∫x∈∂V

n ·[



A(x1, x)

iωρA(x)

]dx

− iω∫

x∈VG∗

A(x1, x)GB(x2, x)�κ(x) dx + 1

iω

∫x∈V

∇G∗A(x1, x) · ∇GB(x2, x)

�ρ(x)

ρA(x)ρB(x)dx

+ 2ω

∫x∈V

G∗A(x1, x)GB(x2, x)ImκA(x) dx =

{−GB(x1, x2), x1 ∈ V0, x1 /∈ V .

(37)

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At first sight it appears that the different choices for the volumes result in different generalized representation theorems. Realizing,however, that the end result should be independent of the choice of the volume, it follows from equating the resulting representations that

GB(x1, x2) = −{

G∗A(x1, x2) + 2ω

∫x∈D

G∗A(x1, x)GB(x2, x)ImκA(x) dx − iω

∫x∈D

G∗A(x1, x)GB(x2, x)�κ(x) dx

+ 1

iω

∫x∈D

∇G∗A(x1, x) · ∇GB(x2, x)

�ρ(x)

ρA(x)ρB(x)dx +

∫x∈∂D

n ·[



A(x1, x)

iωρA(x)

]dx

},

(38)

where I used that V ∪ V ′ = D, ∂V ∪ ∂D = ∂V ′, n = −n′ on ∂V , and n = n′ on ∂D (Fig. 2). This equation also follows immediatelyfrom (35) using D as the integration volume with x1,2 ∈ D, and point sources for the volume injection rate density sources q. Eq. (38) hasthe flavor of the Lippmann–Schwinger eq. (cf. eq. 24) with a subtle difference; it expresses the total wavefield in terms of the time-advanced(i.e. anticausal) Green’s function in the background medium A, while the standard Lippmann–Schwinger equation is expressed in terms ofthe time-retarded (i.e. causal) Green’s function of the background medium. Moreover, it contains a volume integral containing the imaginarycomponent of the background compressibility κA. Since eqs (36) and (37) are equivalent but (36) has the simplest form, I denote (36) as thegeneralized correlation-type acoustic representation theorem for the total wavefield.

It is important to realize that although the boundary integral in the convolution-type generalized theorem (cf. eq. 32) can in principlevanish when the boundary is chosen at infinity, this is never true for the correlation-type equivalent, as the time-advanced Green’s functiondoes not satisfy radiating boundary conditions. This is essentially the foundation of the many recent papers that appeared on the retrieval ofthe Green’s function between two points from the cross-correlation of the passive responses at both locations.

N U M E R I C A L E X A M P L E S

In order to verify the accuracy of the generalized representation theorems for the total wavefield, I consider the simple case of 1-D acousticwave propagation. For the unperturbed state A, I consider a homogeneous medium with a velocity c = 2 km s−1 and density ρ = 1 g cm−3. Instate B, I introduce a localized velocity perturbation (Fig. 4) but leave the density as in the unperturbed medium. The particular perturbationused is one realization of a random fractal. Douma & Roy-Chowdhury (2001) used similar random (albeit binary) fractals to study theinfluence of multiple scattering on the seismic amplitudes in an attempt to use these observations to make an inference about the importanceof multiple scattering in the lower crust as determined from amplitude observations. Fig. 5 shows the reflection response from the medium instate B. The long diffusive tail of energy shows the presence of substantial multiply scattered waves, that is, substantial nonlinear interactionbetween the wavefield and the inhomogeneities.

Now, I turn to the numerical verification of the generalized representation theorems. In the interest of brevity I will focus on theconvolution-type theorem only. For all cases, the boundary-integrals involve the following wavefields:

(i) GA(x1, x) and ∇GA(x1, x),(ii) GB(x2, x) and ∇GB(x2, x),

where x are the source locations on the bounding surface ∂V that (just) encompasses the perturbation. In one dimension, the surface ∂Vmerely reduces to a point above and below the perturbation. That is, we need to calculate the Green’s function GA(x1, x) in medium A fromboth these source locations to x1, as well as the corresponding dipole source responses ∇GA(x1, x). Similarly, we need to calculate GB(x2, x)

Figure 4. Source locations (stars) and receiver locations (triangles) used in the numerical verification of the boundary integrals to calculateGA(x1, x′),∇′GA(x1, x′), GB(x2, x′) and ∇′GB(x2, x′). The perturbation in state B is a random fractal with correlation length a = 15 m, Hurst num-ber ν = 0.3, average velocity c = 2 km s−1, and standard deviation σ = 25 per cent of the average velocity.

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328 H. Douma

Figure 5. Reflection response from the random fractal (Fig. 4, state B) clearly showing the diffusive character of the wavefield due to the presence of substantialmultiply scattered waves. The unscattered direct wave arrives at around 0.2 s.

Figure 6. Verification of the accuracy of the generalized representation theorem for all cases (i)–(iv) depicted in panels (a)–(d), respectively. For all cases thedrawn wavefield represents what the representation theorem should provide, [i.e. G B (x1, x2, t) in cases (i) and (iii), G A(x1, x2, t) in case (ii), and zero in case(iv)], whereas the dotted wavefield represents the response calculated using our generalized representation theorem [i.e. eq. (26) in (a), eq. (27) in (b), eq. (28)in (c) and eq. (30) in (d)]. The differences are indicated by the dashed lines.

and ∇GB(x2, x) from both source locations to x2. To calculate these wavefields I used the propagator-matrix method (Thomson 1950; Haskell1953) for the special case of 1-D propagation in a 1-D acoustic medium. Hence, I used the (numerically) exact GA,B and ∇GA,B to evaluatethe generalized convolution-type representation theorem. The source locations on the bounding surface ∂V (i.e. the source above and belowthe perturbation) are depicted in Fig. 4 by the stars, while the receiver locations x1 and x2 are indicated by the triangles for all four cases.

Fig. 6(a) shows the result of the verification of the generalized convolution-type representation theorem in case (i) for GB(x1, x2, t)in the time-domain. For display purposes all Green’s functions shown throughout the text are convolved with a Ricker wavelet with a peakfrequency of 7.5 Hz. The top response (drawn) is the true response GB(x1, x2, t) calculated with a source in x2 and a receiver in x1, whilethe middle response (dotted) is the response calculated with the boundary-integral representation from eq. (26). The difference between bothseismograms is shown at the bottom (dashed). All responses are shown at the same horizontal and vertical scale. Clearly the generalizedrepresentation theorem allows accurate calculation of GB(x1, x2, t), including all the nonlinear interactions between the wavefield and theinhomogeneities. That is, all multiple scattering is accurately accounted for. Similarly for cases (ii) and (iii), the boundary-integrals of eqs (27)and (28) provide an accurate representation of GA(x1, x2, t) and GB(x1, x2, t), respectively (Figs 6b and c). Finally, Fig. 6(d) verifies that the

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boundary-integral in case (iv) is indeed zero, thus showing that if x1,2 are inside the perturbed area, we cannot use a similar boundary-integralrepresentation as in cases (i) and (iii) to model GB(x1, x2, t). Figs 6(b) and (d) confirm the generalized Ewald–Oseen extinction theorem.The minute differences observed for all four cases can be attributed to the tapering necessary to avoid Gibbs phenomena when numericallyapproximating a Green’s function.

P O T E N T I A L A P P L I C AT I O N S

When the media in state A and B are the same, ignoring attenuation, and assuming x2 ∈ V ′, eq. (36) reduces to∫x∈D

G∗(x1, x)δ(x − x2) dx +∫

x∈∂Dn ·

[G∗(x1, x)∇G(x2, x) − G(x2, x)∇G∗(x1, x)

iωρ(x)

]dx =

{−G(x1, x2), x1 ∈ D0, x1 /∈ D

. (39)

Comparing this to eq. (20), which in the absence of sources in D is known as the integral theorem of Helmholtz and Kirchhoff, we see thatboth equations are the same except that the time-retarded Green’s functions G(x1, x) on the left-hand side of eq. (20) are replaced with thetime-advanced Green’s functions G∗(x1, x) in eq. (39). Korneev & Bakulin (2006) show that eq. (39) is at the basis of the virtual sourcemethod (Bakulin & Calvert 2006) that indeed evaluates the surface integral in eq. (39) by cross-correlating wavefields measured at differentreceivers to find the Green’s function as if one of the receivers was replaced with a source. Hence, one of the receivers acts as a virtual source.Based on this principle each receiver in, for example, a borehole can thus be transformed into a virtual source at depth and can be used, forexample, to image below complex overburden structures such as salt domes (Bakulin et al. 2007) without knowledge of the overburden.

Several authors have shown the application of the virtual source method to time-lapse seismic monitoring by calculating the responsesfrom virtual sources for a base survey and a monitor survey, both based on eq. (39), and comparing the differences (Bakulin & Calvert 2006;Bakulin et al. 2007; Mehta et al. 2008). This approach improves the repeatability of monitoring surveys compared to using conventionalsurveys by eliminating the uncertainties in the source locations. Bakulin & Calvert (2006) show that windowing the response at the virtualsource location around the direct arrival prior to cross-correlation improves the quality of the resulting virtual-source gathers, while Mehtaet al. (2008) show that the effects of seasonal seawater variations can be reduced by incorporating up-down wavefield separation in the cross-correlation process. Eq. (36) suggests an alternative approach by mixing the measured wavefields from the base and monitor surveys in thecross-correlation process. This approach has the advantage that only half of the number of cross-correlations needs to be calculated comparedto the current approach and is likely to benefit equally from the above-mentioned improvements. A study comparing both approaches wouldbe necessary to judge the relative benefits of both methods.

In time-lapse seismic monitoring studies such as monitoring of CO2-sequestration, hydrocarbon reservoirs, or nuclear waste storagesites, the wavefields and their derived images from a reference and monitor survey are compared to infer changes in the medium. In order toestablish the feasibility of detecting and imaging such changes or to determine the optimal timing of monitor surveys, it is necessary to be ableto model the seismic wavefield for models that have local perturbations in the reservoir only. Such modelling needs to be efficient becausethese feasibility studies involve modelling for a whole suite of possible geostatistical models and for different possible production scenarios(Biondi et al. 1998; Robertsson et al. 2000). Several methods exist that allow efficient calculation of approximate wavefields by calculatingonly the wavefield in a subvolume where the medium changed (see, for example, Robertsson & Chapman 2000, and references therein). Suchmethods are generally refered to as wavefield injection methods because the wavefield in the unchanged part of the medium is injected intothe boundary of the subvolume. Such methods ignore any contributions to the wavefield that scatter from the subvolume more than once; inthis way the approximation is similar to the Born approximation. Hence, the gain in efficiency is obtained at the price of ignoring multipleinteractions between inhomogeneities in the reference model and the subvolume. The generalized representation theorems I present do notsuffer from such an approximation and allow calculation of the full Green’s function including all nonlinear interactions. These representationtheorems, however, involve Green’s functions for monopole and dipole sources surrounding the local perturbation calculated for the ‘whole’domain, including the area that did not change.

Often, such as in monitoring of CO2 sequestration or hydrocarbon reservoirs, or maybe even in nuclear-waste storage monitoring, thearea of change is expected to be local. In the cases of CO2 sequestration or hydrocarbon reservoir monitoring, the area of change is expectedto be local due to local injections of CO2 or steam. When the number of sources necessary to ‘cover’ the boundary of the locally perturbedarea is sufficiently smaller than the number of sources that need to be modeled in the acquisition geometry, the generalized representationtheorem of the convolution-type potentially allows efficient calculation of the full Green’s function of the perturbed medium.

Fig. 7 shows a cartoon indicating a possible use of the generalized convolution-type representation theorem in cases (i) and (iii) inthe context of time-lapse monitoring. When the area that is changing is sufficiently small compared to the areal extent of the acquisitiongeometry, the number of sources needed to ‘cover’ the bounding surface ∂V of the perturbed area can be substantially smaller than thenumber of sources in the acquisition geometry. This is often the case in hydrocarbon reservoir monitoring (Fig. 7). In that case it couldwell be beneficial to calculate the Green’s function between any two points in the medium (except when both points are inside the perturbedarea) using the generalized convolution-type representation theorem I present here. The number of forward simulations would then belimited to the number of sources on ∂V instead of doing forward calculations for each source used in the acquisition geometry. Havingsaid that, the calculation of the boundary-integrals involves both forward simulations for monopole and dipole sources, hence doubling theamount of forward simulations. Moreover, the calculation of the Green’s function between two points based on the generalized representationtheorems involves convolutions of G and ∇G integrated over the whole surface ∂V . Therefore, such calculation involves many convolutions.

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330 H. Douma

Figure 7. Cartoon showing the use of the generalized convolution-type representation theorem in cases (i) and (iii) in the context of monitoring a hydrocarbonreservoir. When the localized area where the medium is changing (i.e. the reservoir indicated in grey) is small enough compared to the size of the acquisitionsurvey, the number of sources needed on ∂V to calculate the boundary-integral (crosses) is substantially smaller than the number of sources in the acquisitiongeometry (stars). Then the generalized representation theorem for case (i) and (iii) allows efficient calculation of the full Green’s function in the perturbedmedium. (a) Cross section of the medium showing the surface acquisition geometry with its sources (stars) and receivers (triangles) and possible ‘downhole’receivers in a well drilled into the reservoir. (b) Plan view of source distribution in typical acquisition geometry, showing that if the perturbation is smallenough, indeed much fewer sources are needed to ‘cover’ the boundary of the perturbed area (crosses) compared to the number of sources in the acquisitiongeometry (stars).

Hence, even though in principle fewer forward simulations need to be calculated, this reduction is achieved at the cost of having tocalculate many convolutions. Therefore, it seems that numerical efficiency will be gained only when the number of sources needed tocover the bounding surface ∂V is indeed much smaller than the number of sources in the acquisition geometry. We emphasize here theapplication of the convolution-type generalized representation theorem because the boundary integral over ∂D vanishes due to radiating(or free) boundary conditions, while for the correlation-type theorem this integral (remember that ∂V ′ = ∂D ∪ ∂V) does in general notvanish.

Even though I mention the use of the generalized convolution-type represenation theorem for cases (i) and (iii) in time-lapse seismicstudies, this theorem can also potentially be useful in non-linear inversions. Usually a non-linear inverse problem is solved by repeatedlysolving a linear inversion; that is, the model is updated using a linear inversion step, and subsequently this model is used as the backgroundmodel for another linear inversion step. In this way one hopes to converge to the solution that is optimal in some sense. The generalizedconvolution-type representation theorem for cases (i) and (iii) can be potentially beneficial for efficient wavefield modelling, provided againthat an update is done only locally and that the number of sources needed to ‘cover’ the boundary of the perturbed area is much smaller thanthe number of sources for which the Green’s function needs to be known. Likely there are other applications in non-destructive testing that arebeyond the author’s current imagination but would benefit from the presented generalized representation theorems for the total wavefield.

D I S C U S S I O N

Although the numerical validation in one dimension clearly showed the accuracy of the generalized convolution-type representation theorem,there is a possible caveat in higher dimensions. In the latter case, the wavefield usually contains turning waves that are inhomogeneous belowa certain depth. In principle, these waves are obviously excited by the monopole and dipole sources on the boundary of the perturbed area ∂V .From a computational point of view, however, the excitation of the inhomogeneous waves, which are exponentially decreasing with depth,involves exponentially increasing amplitudes. This could well lead to instabilities. This becomes particularly important when both points x1,2

are outside of the perturbation area and substantially above the depth where certain waves become inhomogeneous. It remains to be seen inhow far this caveat forms an obstacle in any practical applications.

In this paper, I treat acoustic media only. In light of the recent work by Wapenaar et al. (2006) and Snieder et al. (2007), however,the presented derivations apply also to the more general cases of elastic wave propagation in attenuative as well as non-attenuative media,electromagnetic wave propagation in conducting media, Green’s function retrieval of diffuse wavefields, and even to the estimation of theelectrokinetic Green’s function in piezoelectric or poroelastic media.

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C O N C LU S I O N

Based on reciprocity theorems of the convolution- and correlation-type for perturbed media, I have derived generalized convolution- andcorrelation-type representation theorems for the total acoustic wavefield in perturbed media. These theorems allow calculation of the totalwavefields in perturbed media by evaluating a boundary integral that requires knowledge of the impulse responses of monopole and dipolesources at the boundary only.

The derivation using the convolution-type reciprocity theorem makes clear the close connection between this theorem and the Lippmann–Schwinger equation for the total wavefield. When the receiver location is in the scattering volume, the generalized representation theorem ofthe convolution-type specializes to a generalization of the Ewald–Oseen extinction theorem. Based on the principle that the integration volumein the reciprocity theorems can be chosen freely and that different choices of this volume should lead to the same resulting representationtheorem, I show that the correlation-type reciprocity theorem leads to a new Lippmann–Schwinger type equation for the total wavefield butin terms of the time-advanced instead of the time-retarded background wavefield.

In case the perturbation is zero everywhere, the generalized correlation-type representation theorem simplifies to the basic equationunderlying the virtual source method. Therefore, this generalized representation theorem is applicable to the virtual-source method in thecontext of time-lapse seismics, and in particular provides an alternative way to calculate virtual-source difference wavefields compared to thecurrent practice of differencing virtual-source wavefields from the base and monitor surveys. In addition, I have suggested the possible useof the convolution-type representation theorem in the context of efficient modelling of wavefields in time-lapse seismics.

A C K N OW L E D G M E N T S

I would like to thank Tony Dahlen, Guust Nolet and Roel Snieder for several useful discussions, their encouragement and support, but mostlyfor their inspiration. The careful reviews of Andrey Bakulin, Kees Wapenaar, Jacob Fokkema and Ivan Vasconcelos that helped improve theoriginal manuscript are much appreciated. Financial support for this study has been provided by the US National Science Foundation undergrants DMS-0530865, EAR-0309298 and EAR-0105387.

R E F E R E N C E S

Aki, K. & Richards, P.G., 2002. Quantitative Seismology, 2nd edn, Univer-sity Science Books, Sausalito, California.

Baker, B. & Copson, E., 1950. The Mathematical Theory of Huygens’ Prin-ciple, Clarendon Press, Oxford.

Bakulin, A. & Calvert, R., 2006. The virtual source method: theory and casestudy, Geophysics, 71, SI139–SI150.

Bakulin, A., Mateeva, A., Mehta, K., Jorgensen, P., Ferrandis, J., SinHaHerhold, I. & Lopez, J., 2007. Virtual source applications to imaging andreservoir monitoring, Leading Edge, 26, 732–740.

Biondi, B., Mavko, G., Mukerji, T., Rickett, J., Lumley, D., Deutsch, C.,Gundeso, R. & Thiele, M., 1998. Reservoir monitoring: a multidisci-plinary feasibility study, Leading Edge, 17, 1404–1414.

Borcea, L., Papanicolaou, G. & Tsogka, C., 2005. Interferometric arrayimaging in clutter, Inverse Problems, 21, 149–1460.

Born, M. & Wolf, E., 1970. Principles of Optics, 3rd edn, Pergamon Press,Oxford.

Campillo, M. & Paul, A., 2003. Long-range correlations in the diffuse seis-mic coda, Science, 299(5606), 547–549.

Cerveny, V., 2001. Seismic Ray Theory, Cambridge University Press,Cambridge.

Claerbout, J., 1968. Synthesis of a layered medium from its acoustic trans-mission response, Geophysics, 33(2), 264–269.

de Goede, J. & Mazur, P., 1972. On the extinction theorem in electrodynam-ics, Physica, 58, 568–584.

de Hoop, A.T., 1988. Time-domain reciprocity theorems for acoustic wavefields in fluids with relaxation, J. acoust. Soc. Am, 84, 1877–1882.

Dillen, M., Fokkema, J. & Wapenaar, C., 2002. Recursive elimination of tem-poral contrasts between time-lapse acoustic wave fields, J. Seism. Explor.,11, 41–58.

Douma, H. & Roy-Chowdhury, K., 2001. Amplitude effects due tomulti-scale impedance contrasts and multiple scattering: implicationsfor IVREA-type continental lower crust, Geophys. J. Int., 147, 435–448.

Duvall, T.L., Jefferies, S.M., W., H.J. & Pomerantz, 1993. Time distancehelioseismology, Nature, 362, 430–432.

Ewald, P., 1916. Zur Begrundung der Kristalloptik (english translation: onthe foundation of crystal optics), Annalen der Physik, Series 4, 49, 1–38,117–143.

Fearn, H., James, D. & Milonni, P., 1996. Microscopic approach to reflec-tion, transmission, and the ewald-oseen extinction theorem, Am. J. Phys.,64, 986–995.

Fokkema, J.T. & van den Berg, P.M., 1993. Seismic Applications of AcousticReciprocity, Elsevier, Amsterdam.

Haskell, N., 1953. The dispersion of surface waves on multilayered media,Bull. seism. Soc. Am., 43, 17–34.

Korneev, V. & Bakulin, A., 2006. On the fundamentals of the virtual sourcemethod, Geophysics, 71, A13–A17.

Maradudin, A., Mendez, E. & Michel, R., 1989. Backscattering effects inthe elastic scattering of p-polarized light from a large-amplitude randommetallic grating, Optics Lett., 14, 151–153.

Mehta, K., Sheiman, J., Snieder, R. & Calvert, R., 2008. Strengthening thevirtual-source method for time-lapse monitoring, Geophysics, 73, S73–S80.

Morse, P. & Feshbach, H., 1953. Methods of Theoretical Physics,Vol. II,McGraw-Hill, New York.

Oseen, G., 1915. Uber die Wechselwirkung zwischen zwei elektrischenDipolen und uber die Drehung der Polarisationsebene in Kristallen undFlussigkeiten, (english translation: on the interaction between two elec-tric dipoles and on the rotation of the polarization-plane in crystals andfluids), Annalen der Physik, 48, 1–56.

Pattanayak, D. & Wolf, E., 1972. General form and a new interpretation ofthe Ewald-Oseen extinction theorem, Opt. Commun., 6(3), 217–220.

Rayleigh, 1945. Theory of Sound,Vol. II, Dover Publications, New York.Rickett, J. & Claerbout, J., 1999. Acoustic daylight imaging via spectral

factorization: helioseismology and reservoir monitoring, Leading Edge,18, 957–960.

Rickett, J. & Claerbout, J., 2000. Calculation of the sun’s acoustic impulseresponse by multi-dimensional spectral factorization, Sol. Phys., 192,203–210.

Robertsson, J. & Chapman, C., 2000. An efficient method for calculatingfinite-difference seismograms after model alterations, Geophysics, 65(3),907–918.

C© 2009 The Author, GJI, 179, 319–332


Dow



332 H. Douma

Robertsson, J., Ryan-Grigor, S., Sayers, C. & Chapman, C., 2000. A finite-difference injection approach to modeling seismic fluid flow monitoring,Geophysics, 65(3), 896–906.

Sabra, K., Gerstoft, P., Roux, P., Kuperman, W. & Fehler, M., 2005. Surfacewave tomography from microseims in southern california, Geophys. Res.Lett., 32, L14311, doi:10.1029/2005GL023155.

Sanchez-Gil, J. & Garcıa-Ramos, J., 1998. Calculations of the direct electro-magnetic enhancement in surface enhanced Raman scattering on randomself-affine fractal metal surfaces, J. Chem. Phys, 108, 317–325.

Sanchez-Gil, J. & Nieto-Vesperinas, M., 1991. Light scattering from randomrough dielectric surfaces, J. Opt. Soc. Am., 8, 1270–1286.

Schiff, L., 1968. Quantum Mechanics, 3rd edn, McGraw-Hill Inc., NewYork, NY.

Schuster, G., 2001. Theory of daylight/interferometric imaging: tutorial,session a32, in 63rd Annual Meeting of the European Association ofGeoscientists and Engineers, Extended Abstracts, EAGE.

Schuster, G., Yu, J., Sheng, J. & Rickett, J., 2004. Interferometric/daylightseismic imaging, Geophys. J. Int., 157, 838–852.

Sein, J., 1970. A note on the Ewald-Oseen extinction theorem, Opt. Com-mun., 2(4), 170–172.

Sein, J., 1975. General extinction theorems, Opt. Commun., 14(2), 157–160.

Sens-Schonfelder, C. & Wegler, U., 2006. Passive image interferometry andseasonal variations of seismic velocities at Merapi volcano, Indonesia,Geophys. Res. Let., 33, L21302, doi:10.1029/2006GL027797.

Shapiro, N.M., Campillo, M., Stehly, L. & Ritzwoller, M.H., 2005. High-Resolution Surface-Wave Tomography from Ambient Seismic Noise, Sci-ence, 307(5715), 1615–1618.

Slob, E. & Wapenaar, K., 2007. Electromagnetic Green’s’s functions re-trieval by cross-correlation and cross-convolution in media with losses,Geophys. Res. Lett., 34, L05307, doi:10.1029/2006GL029097.

Snieder, R., 2004. Extracting the Green’s’s function from the correlationof coda waves: a derivation based on stationary phase, Phys. Rev. E, 69,046610-1–046610-8.

Snieder, R., 2007. Extracting the Green’s’s function of attenuating heteroge-neous acoustic media from uncorrelated waves, J. acoust. Soc. Am., 121,2637–2693.

Snieder, R. & Safak, E., 2006. Extracting the building response using seismic

interferometry; theory and application to the millikan library in pasadena,california, Bull. seism. Soc. Am., 96, 586–598.

Snieder, R., Wapenaar, K. & Wegler, U., 2007. Unified Green’s’s functionretrieval by cross-correlation; connection with energy principles, Phys.Rev. E, 75, 036103.

Stehly, L., Campillo, M. & Shapiro, N., 2006. A study of the seismic noisefrom its long-range correlation properties, J. geophys. Res., 111, B10306,doi:10.1029/2005JB004237.

Thomson, W., 1950. Transmission of elastic waves through a stratifiedmedium, J. Appl. Phys., 21, 89–93.

van Manen, D.-J., Robertsson, J.O.A. & Curtis, A., 2005. Modeling of wavepropagation in inhomogeneous media, Phys. Rev. Lett., 94(16), 164301.

van Manen, D.-J., Curtis, A. & Robertsson, J.O.A., 2006. Interferometricmodeling of wave propagation in inhomogeneous elastic media usingtime reversal and reciprocity, Geophysics, 71(4), SI47–SI60.

Vasconcelos, I., 2007. Interferometry in perturbed media: Reciprocity The-orems, Deconvolution Interferometry, and Imaging of Borehole Seis-mic Data, PhD thesis, Colorado School of Mines, Center for WavePhenomena.

Vasconcelos, I., Snieder, R. & Hornby, B., 2008. Imaging internal multi-ples from subsalt VSP data examples of target-oriented interferometry,Geophysics, 73, S157–S168.

Wapenaar, K., 2004. Retrieving the elastodynamic Green’s’s function of anarbitrary inhomogeneous medium by cross correlation, Phys. Rev. Lett.,93(25), 254301.

Wapenaar, K., Dillen, M. & Fokkema, J., 2001. Applying one-way reci-procity theorems in time-lapse seismic imaging, J. Seism. Explor., 10,165–181.

Wapenaar, K., Slob, E. & Snieder, R., 2006. Unified Green’s’s functionretrieval by cross correlation, Phys. Rev. Lett., 97, 234301-1–234301-4.

Weaver, R.L. & Lobkis, O.I., 2001. Ultrasonics without a source: Ther-mal fluctuation correlations at mhz frequencies, Phys. Rev. Lett., 87,134301.

Xiao, X., Zhou, M. & Schuster, G., 2006. Salt-flank delineation by in-terferometric imaging of transmitted P- to S-waves, Geophysics, 71(4),SI197–SI207.

Yilmaz, O., 2001. Seismic Data Analysis,Vol. II, Society of ExplorationGeophysics.

C© 2009 The Author, GJI, 179, 319–332


Dow



generalized representation theorems for acoustic wavefields in

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