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Representation theorems in perturbed acoustic media:

Applications to interferometry and imaging

Ivan Vasconcelos1,2

and Roel Snieder2

1ION Geophysical, GXT Imaging Solutions, Egham, Surrey, UK2Center for Wave Phenomena, Colorado School of Mines

ABSTRACT

General reciprocity theorems for perturbed acoustic media are provided in theform of convolution- and correlation-type theorems. The results presented herediffer from previous derivations because they provide explicit integral relationsbetween wavefields and wavefield perturbations which alone do not satisfy theacoustic wave equations. Using Green’s functions to describe perturbed and un-perturbed waves in two distinct wave states, representation theorems are derivedfrom the reciprocity relations in perturbed media. While the convolution-typetheorems can be manipulated to obtain the Lippmann-Schwinger integral, thecorrelation-type theorems can be used to retrieve the perturbation response ofthe medium by cross-correlations. By cross-correlating wavefield perturbationsrecorded at a given receiver with unperturbed waves at another, one obtainsa pseudo-experiment where only wavefield perturbations propagate from onereceiver to the other as if one of them were a source. This can be done whenthe pressure fluctuations arise from random volume sources; or from coherentexcitation by surface sources when the receivers lie away from the perturbedportion of the medium. The perturbation-based integral equations presentedhere are particularly important for the description of scattering problems, andfor reconstructing only field perturbations (e.g., scattered waves) from cross-correlating measured field quantities.

This paper has been submitted to Physical Review E.

Key words: 3D, 9C, wide-azimuth seismic data, fracture characterization,multiple fracture sets, orthorhombic anisotropy

1 INTRODUCTION

Reciprocity theorems have long been used to describeimportant properties of wave propagation phenomena.Lord Rayleigh (Rayleigh, 1878) used a local form ofan acoustic reciprocity theorem to demonstrate source-receiver reciprocity. Time-domain reciprocity theoremswere later generalized to relate two wave states with dif-ferent field, material and source properties in absorbing,heterogeneous media (de Hoop, 1988).

Fokkema and van den Berg (1993) show that acous-tic reciprocity theorems can be used for modeling wavepropagation, for boundary and domain imaging, andfor estimation of the medium properties. In the field

of exploration seismology, an important application ofconvolution-type reciprocity theorems lies in removingmultiple reflections, also called multiples, caused by theEarth’s free-surface (Fokkema & van den Berg, 1993;Berkhout & Verschuur, 1997). These approaches rely onthe convolution of single-scattered waves to create mul-tiples, which are then adaptively subtracted from therecorded data. Other approaches for the elimination ofmultiples from seismic data rely on inverse scatteringmethods (Weglein et al., 2003; Malcolm et al., 2004).The inverse-scattering based methodologies are typi-cally used separately from the representation theoremapproaches (Fokkema & van den Berg, 1993; Berkhout& Vershuur, 1997) in predicting multiples. Note: Rep-

282 I. Vasconcelos & R. Snieder

resentation theorems are derived from reciprocity theo-rems using Green’s functions; e.g., see Section 3 of thispaper.

Recent forms of reciprocity theorems have been de-rived for the extraction of Green’s functions (Wape-naar et al., 2002; Wapenaar & Fokkema, 2006; Wape-naar et al., 2006), showing that the cross-correlations ofwaves recorded by two receivers can be used to obtainthe waves that propagate between these receivers as ifone of them behaves as a source. These results coincidewith other studies based on cross-correlations of diffusewaves in a medium with an irregular boundary (Lobkis& Weaver, 2001), caused by randomly distributed un-correlated sources (Weaver & Lobkis, 2001; Wapenaaret al., 2002; Shapiro et al., 2005), or present in the codaof the recorded signals (26). (Snieder, 2004). An earlyanalysis by Claerbout (1968) shows that the reflectionresponse in a 1D medium can be reconstructed fromthe autocorrelation of recorded transmission responses.This result was later extended for cross-correlations inheterogeneous 3D media by Wapenaar et al. (2004),who used one-way reciprocity theorems in their deriva-tions. Green’s function retrieval by cross-correlationshas found applications in the fields of global (Shapiroet al., 2005; Sabra et al., 2005) and exploration seismol-ogy (Schuster et al., 2004; Willis et al., 2006; Bakulin& Calvert, 2006), ultrasonics (Malcolm et al., 2004;wan Wijk 2006), helioseismology (Rickett & Claerbout,1999), structural engineering (Snieder & Safak, 2006;Thompson & Snieder, 2007) and ocean acoustics (Rouxet al., 2004; Sabra et al., 2005).

Although the correlation-based Green’s function re-trieval has been proven for special cases by methodsother than reciprocity theorems (e.g., [Lobkis et al.,2001; Bakulin & Calvert, 2006]), the derivations basedon reciprocity theorems have provided for generaliza-tions beyond lossless acoustic wave propagation to elas-tic wave propagation and diffusion (40; 27). (Wapenaar,2004; Snieder, 2006). More general forms of reciprocitytheorems have been derived (Wapenaar et al., 2006;Snieder et al., 2007; Weaver, 2008) which include a widerange of differential equations such as the acoustic, elas-todynamic, and electromagnetic wave equations, as wellas the diffusion, advection and Schrodinger equations,among others.

In this paper, we derive reciprocity theorems foracoustic perturbed media. The perturbations of thewavefield due to the perturbation of the medium canbe used for imaging or for monitoring. For imaging, theunperturbed medium is assumed to be so smooth thatis does not generate reflected waves, while the pertur-bation accounts for the reflections. In monitoring ap-plication, the perturbation consists of the time-lapsechanges in the medium. Although previous derivationsof reciprocity theorems account for arbitrary mediumparameters that are different between two wave states(de Hoop, 1988; Fokkema & van den Berg, 1993), they

do not explicitly consider the special case of perturbedmedia. In perturbed media, there are special relationsbetween the unperturbed and perturbed wave states(e.g., in terms of the physical excitation) that make thereciprocity theorems in such media differ in form withrespect to their more general counterparts (Fokkema &van den Berg, 1993; Wapenaar et al., 2006). We discusssome of these differences. Another important aspect ofstudying representation theorems in perturbed medialies in retrieving wavefield perturbations from cross-correlations, in a manner analogous to that presentedearlier (Wapenaar et al., 2006; Snieder, 2007). As weshow here, wavefield perturbations by themselves do notsatisfy the wave equations and thus their retrieval doesnot follow directly from these earlier derivation. In par-allel with the work we present here, Douma (2007) de-rived convolution- and correlation-type reciprocity the-orems for acoustic perturbed media that are related tothe Lippmann-Schwinger series, with the purpose of ac-complishing time-lapse monitoring by non-linear inver-sion.

We first derive general forms of convolution- andcorrelation-type reciprocity theorems by manipulatingthe perturbed and unperturbed wave equations for twowave states. In the Section that follows, we write rep-resentation theorems using the Green’s functions forunperturbed and perturbed waves in the two states.We show that the convolution-type theorems result inscattering-like integrals that can be used to describethe field perturbations between two observation points.Next we analyze how the correlation-type theoremscan be used to extract the field perturbations fromcross-correlations of observed fields, for different caseof medium and experiment configurations. Finally, wediscuss the applications of these representation theo-rems in recovering the perturbation response betweentwo sensors from random medium fluctuations and fromcoherent surface sources; and also their applications intomography.

2 RECIPROCITY THEOREMS IN

CONVOLUTION AND CORRELATION

FORM

We define acoustic wave states in a domain V, boundedby ∂V (Figure 1). The outward pointing vector normalto ∂V is represented by n. We consider two wave states,which we denote by the superscripts A and B, respec-tively. Each wave state is defined in an unperturbedmedium with compressibility κ0(r) and density ρ0(r); aswell as in a perturbed medium described by the materialproperties κ(r) and ρ(r). Using the Fourier conventionu(t) =

R

u(ω) exp(−iωt)dω, the acoustic wavefield equa-tions for state A in an unperturbed medium are, in the

Representation theorems in perturbed acoustic media 283

Figure 1. Illustration of the domain used in the reciprocitytheorems. The domain consists of a volume V, bounded by∂V. The unit vector normal to ∂V is represented by n. Thewave states A and B are represented by receivers placedat rA (white triangle) and rB (grey triangle), respectively.The solid arrows denote the stationary paths of unperturbedwaves G0, propagating between the receivers and an arbi-trary point r on ∂V.

frequency-domain,

∇pA0 (r, ω) − iωρ0(r)v

A0 (r, ω) = 0

∇ · vA0 (r, ω) − iωκ0(r)p

A0 (r, ω) = qA(r, ω) ,

(1)

where pA(r, ω) and vA(r, ω) represent pressure and par-ticle velocity, respectively. The quantity qA(r, ω) de-scribes the source distribution as a volume injection ratedensity. Similar equations describe waves in state A ina perturbed medium:

∇pA(r, ω) − iωρ(r)vA(r, ω) = 0

∇ · vA(r, ω) − iωκ(r)pA(r, ω) = qA(r, ω) ,(2)

with perturbed pressure and particle velocity given bypA(r, ω) and vA(r, ω), respectively. The acoustic fieldequations that describe wave propagation in state B fol-low by replacing the superscript A by B in equations 2and 1. Note that the source distribution qA(r, ω) is thesame for both the unperturbed (equation 1) and per-turbed (equation 2) cases. We assume that no exter-nal volume forces are present by setting the right-sideof the vector equations in equations 1 and 2 equal tozero. The perturbed pressure for either wave state isgiven by p = p0 + pS, where the subscript S indicatesthe wavefield perturbation caused by medium changes.For brevity, we assume that perturbations only occur incompressibility, thus ρ = ρ0, but our derivations can begeneralized to include density perturbation as well. Notethat there are no restrictions about the smoothness ofthe material parameters, i.e., rapid lateral changes anddiscontinuities are allowed.

To derive Rayleigh’s reciprocity theorem (Rayleigh,1878; de Hoop, 1988; Fokkema & van den Berg, 1993) weinsert the equations of motion and stress-strain relationsfor states A and B (equations 1 and 2) in

vB0 · EA

0 + pA0 E

B0 − v

A0 · EB

0 − pB0 E

A0 ; (3)

where E and E represent the equation of motion andthe stress-strain relation in equation 1, respectively. Thesubscripts in equation 3 indicate that equations andfield parameters (p and v) are considered in the unper-turbed case. The superscripts indicate whether equa-tions and field parameters pertain to state A or B. Forbrevity, we omit the parameter dependence on r andω. From equation 3 we isolate the interaction quantity

∇· (pA0 vB

0 − pB0 vA

0 ) (de Hoop, 1988). Next, we integratethe result of equation 3 over the domain V and applyGauss’ divergence theorem. This results inI

r∈∂V

h

pA0 v

B0 − pB

0 vA0

i

·dS =

Z

r∈V

h

pA0 q

B0 − pB

0 qA0

i

dV ;

(4)which is referred to as a reciprocity theorem of the con-volution type (de Hoop, 1988; Fokkema & van den Berg,1993), because the frequency-domain products of fieldparameters represent convolutions in the time domain.A correlation-type reciprocity theorem (de Hoop, 1988;Fokkema & van den Berg, 1993) can be derived fromisolating the interaction quantity ∇ · (pA

0 vB∗0 + pB∗

0 vA0 )

from

vB∗0 · EA

0 + pA0 E

B∗0 + v

A0 · EB∗

0 + pB∗0 EA

0 , (5)

where ∗ denotes complex conjugation. Subsequent vol-ume integration and application of the divergence the-orem yieldsI

r∈∂V

h

pA0 v

B∗0 + pB∗

0 vA0

i

·dS =

Z

r∈V

h

pA0 q

B∗0 + pB∗

0 qA0

i

dV ,

(6)where complex conjugates represents time-domaincross-correlations of field parameters. For this reason,equation 6 is a reciprocity theorem of the correlationtype (de Hoop, 1988; Fokkema & van den Berg, 1993).Convolution- and correlation-type reciprocity theoremsfor the perturbed wave states (e.g., equation 2) can beexpressed simply by removing the subscript 0 from equa-tions 3 though 6. In equation 6 we assume that κ0 andρ0 are real quantities (i.e.; the medium is lossless). Inthe next Section we review the application of such the-orems to retrieving the Green’s functions of arbitrarymedia (Wapenaar et al., 2002; Wapenaar & Fokkema,2006).

The theorems in equations 4 and 6 also hold whenthe material properties in states A and B are the same.General reciprocity theorems that account for arbitrar-ily different source and material properties between twowave states have been derived (de Hoop, 1988; Fokkema& van den Berg, 1993).

Here, we further develop these reciprocity theorems


for the special case of perturbed acoustic media. First,we consider

vB0 · EA + pAEB

0 − vA · EB

0 − pB0 E

A , (7)

which involves the equations and field parameters forstate B in an unperturbed medium (e.g., equation 1),along with equations and field parameters for state A ina perturbed medium (equation 2). From equation 7 weisolate the interaction quantity ∇·(pAvB

0 −pB0 vA). After

separating this quantity, we integrate equation 7 over V

and apply Gauss’ theorem. Next, given that p = p0 +pS

and v = v0 +vS, we use the result in equation 4, whichgives

I

r∈∂V

h

pAS v

B0 − pB

0 vAS

i

· dS

=

Z

r∈V

pAS q

B0 dV +

Z

r∈V

iω(κ0 − κ)pApB0 dV , (8)

which is a convolution-type reciprocity theorem for per-turbed media. This expression is a new form of reci-procity theorem because it relates the wavefields pB

0 andvB

0 with the wavefield perturbations pAS and vA

S . Pre-viously derived reciprocity theorems (de Hoop, 1988;Fokkema & van den Berg, 1993; Wapenaar & Fokkema,2006) provide, for two arbitrary wave states, relationsbetween field parameters that satisfy wave equations(e.g., equations 1 and 2). Wavefield perturbations such aspA

S and vAS , by themselves, do not satisfy the wave equa-

tions for perturbed media (e.g., equation 2). Althoughequation 8 accounts for compressibility changes only, itcan be modified to include density perturbations. Suchmodification involves adding, to the right-hand side ofthe equation, a third volume integral whose integrandis proportional to (ρ0 − ρ) and the wavefields vA andvB

0 (Fokkema & van den Berg, 1993).The correlation-type counterpart of equation 8 can

be derived from the interaction quantity ∇ · (pAvB∗0 +

pB∗0 vA), which can be isolated from

vB∗0 · EA + pAEB∗

0 + vA · EB∗

0 + pB∗0 EA . (9)

After performing the same steps as in the derivation ofequation 8 we obtain

I

r∈∂V

h

pAS v

B∗0 + pB∗

0 vAS

i

· dS

=

Z

r∈V

pAS q

B∗0 dV −

Z

r∈V

iω(κ0 − κ)pApB∗0 dV , (10)

which is a correlation-type reciprocity theorem for per-turbed acoustic media. Again, we assume that both κand κ0 are real (i.e., no attenuation). As with its con-volution counterpart (equation 8), equation 10 is novelbecause it provides a relation between the wavefieldperturbations in state A and the unperturbed wavesin state B. Density perturbations can be included inequation 10 in a manner analogous to that discussed forequation 8 (de Hoop, 1988; Fokkema & van den Berg,1993). By interchanging the superscripts in equations 7

and 9 we derive convolution- and correlation-type reci-procity theorems that relate the perturbations pB

S andvB

S to pA0 and vA

0 . These theorems have the same form asthe ones in equations 8 and 10, except A is interchangedwith B in equation 8, and with B∗ in equation 10.

Since the source term qA(r, ω) is the same both inthe unperturbed and perturbed cases, it follows fromequations 1 and 2 that

∇pA0 (r, ω) − iωρ0(r)v

A0 (r, ω)

= ∇pA(r, ω) − iωρ(r)vA(r, ω)

∇ · vA0 (r, ω) − iωκ0(r)p

A0 (r, ω)

= ∇ · vA(r, ω) − iωκ(r)pA(r, ω) . (11)

These relations can be used to derive others reciprocitytheorems in perturbed media. To derive a convolution-type theorem, we first consider the combination

vB0 · RA + pA

0 RB −vA

0 · RB − pB0 R

A

+vB · RA + pARB −vA · RB − pBRA ,

(12)

where R and R are the vector and scalar relations inequation 11; the superscripts indicate whether they per-tain to wave state A or B. Equation 12 is then integratedover volume. With Gauss’ theorem, the resulting equa-tion can be simplified by using the identities p = p0+pS

and v = v0+vS. In simplifying equation 12, we also usethe convolution-type reciprocity theorem in equation 8as shown, and with interchanged A and B superscripts.These steps result in

I

r∈∂V

h

pAS v

BS − pB

S vAS

i

· dS

=

Z

r∈V

iω(κ− κ0)h

pASp

B0 − pA

0 pBS

i

dV . (13)

Because the frequency-domain products in the inte-grands correspond to convolutions in the time domain,this integral theorem is of the convolution type. Theconvolution-type reciprocity theorem in equation 13 re-lates wavefield perturbations in both wave states Aand B over the surface ∂V with wavefield perturba-tions, unperturbed waves, and the medium perturba-tion (κ− κ0) within the volume V. Equation 13 can beextended to include density perturbations by adding aextra volume integral with the integrand proportionalto (ρ− ρ0)

ˆ

vAS vB

0 − vA0 vB

S

˜

(Fokkema & van den Berg,1993).

A correlation-type theorem of the same form asequation 13 can be derived from modifying the rela-tions in equation 12 to account for the time-reversedwave state B∗ (e.g., equations 3 and 5). Again, we inte-grate the result over V, use Gauss’ theorem, and simplifyit by representing perturbed wavefields in terms of un-perturbed waves and wavefield perturbations. Using thetheorem in equation 10 as is, and with A interchanged


with B∗, we obtainI

r∈∂V

h

pAS v

B∗S + pB∗

S vAS

i

· dS

=

Z

r∈V

iω(κ− κ0)h

pA0 p

B∗S − pA

SpB∗0

i

dV , (14)

which is the correlation-type counterpart of the theo-rem in equation 13. This theorem can be extended to in-clude small density perturbations in a manner analogousto that described for equation 13. Like the reciprocitytheorems in equations 8 and 10, equations 13 and 14provide relations between wavefield perturbations andwavefields that satisfy the unperturbed acoustic waveequation.

3 REPRESENTATION THEOREMS IN

PERTURBED MEDIA

In this Section, we use Green’s functions to derive rep-resentation theorems from the reciprocity theorems inthe previous Section. We focus the discussion on therole of Green’s functions in the reciprocity theorems inequations 10 and 14 because of the applicability of thesetheorems to remote sensing experiments (treated in thenext Section). The Green’s function forms of the theo-rems in equations 4 and 6 are treated by others (Wape-naar et al., 2002; Wapenaar & Fokkema, 2006; Draganovet al., 2006). The convolution theorem in equation 4leads to the well-known acoustic source-receiver reci-procity relation (Rayleigh, 1878; Fokkema & van denBerg, 1993; Wapenaar & Fokkema, 2006). From thecorrelation-type theorem in equation 6, it follows thatthe surface integration of the cross-correlated Green’sfunction results in the causal and anticausal Green’sfunctions that propagate between two receivers (Wape-naar, 2004; Wapenaar & Fokkema, 2006). We discussthis in in more detail below.

We introduce the Green’s functions, in the fre-quency domain, by setting

qA,B = δ(r− rA,B) , (15)

where the positions rA,B denote the wave states A andB, respectively. This choice for q allows for expressingthe field quantity p in terms of the Green’s functions G,i.e.,

pA,B(r, ω) = G(r, rA,B , ω)

= G0(r, rA,B , ω) +GS(r, rA,B , ω) , (16)

where the subscripts 0 and S stand for unperturbedwaves and wavefield perturbations, respectively. Notethat these are the Green’s functions for sources ofthe volume injection rate type. The derivation belowcan also be reproduced using volume injection sources(Wapenaar & Fokkema, 2006). It follows from equa-tions 16 and 2 that vA,B(r, ω) = (iωρ)−1∇G(r, rA,B , ω).

Let us first analyze the Green’s form of the convolution-type theorem in equation 8, by using the definitions inequations 15 and 16. For the moment we assume ho-mogeneous boundary conditions at ∂V that make thesurface integral vanish. These conditions are satisfiedwhen i) a radiation boundary condition holds when thesurface extends to infinity, ii) or the pressures GA,B arezero at ∂V and/or when iii) ∇GA,B ·n = 0. Under theseconditions we obtain

GS(rB , rA) =

Z

r∈V

1

iωρG(r, rA)V (r)G0(r, rB) dV ;

(17)where V (r) = ω2ρ(κ(r) − κ0(r)) is the perturbation

operator or scattering potential (Rodberg & Thaler,1967). For brevity we omit the dependence on the fre-quency ω. Equation 17 is the integral equation known asthe Lippmann-Schwinger equation (Rodberg & Thaler,1967), commonly used in scattering problems. Whennone of the surface boundary conditions listed aboveapply, the surface integral of equation 8 is present onthe right-hand side of equation 17.

Next, we turn our attention to the correlation-typereciprocity theorem in equation 10. Substituting theGreen’s functions (equation 16) for the wavefields p andv in equation 10 gives

Z

r∈V

GS(r, rA)δ(r− rB)dV

=

I

r∈∂V

1

iωρ[GS(r, rA)∇G∗

0(r, rB)

+ G∗0(r, rB)∇GS(r, rA)] · dS

+

Z

r∈V

iω(κ0 − κ)G(r, rA)G∗0(r, rB)dV ; (18)

or simply,

GS(rB , rA)

=

I

r∈∂V

1


0(r, rB)

+ G∗0(r, rB)∇GS(r, rA)] · dS

+

Z

r∈V

iω(κ0 − κ)G0(r, rA)G∗0(r, rB)dV

+

Z

r∈V

iω(κ0 − κ)GS(r, rA)G∗0(r, rB)dV . (19)

The left-hand side describes causal wavefield perturba-tions that propagate from rB to rA as if the observa-tion point at rB acts as a source. This equation showsthat the causal wavefield perturbation GS(rB , rA) is ob-tained from the surface integral of the cross-correlationof wavefield perturbations at rA with unperturbedwaves at rB . Along with this surface integral, the vol-ume integrals in equation 19 are, in general, necessaryto recover GS(rB , rA). By taking equation 19 with in-terchanged superscripts A by B∗, and using reciprocity


(a) (b)

Figure 2. Schematic illustrations of the role of the volume integrals in reconstructing GS(r, rA) for Case I. Medium per-turbations are restricted to the volume P, placed away from the observation points. The solid arrows represent the paths ofunperturbed waves in G0(r, rA,B), while the dotted arrows illustrate the paths of the field perturbations GS(r, rA). For some

sources on ∂V the medium perturbation in P does not affect the path of the measured unperturbed waves, as in the example ofthe source position r1 in panel (a). In contrast, the path of the observed unperturbed waves is affected by P for source positionssuch as r2 in panel (b). Here, the source positions r1 in (a) and r2 in (b) are stationary points for the integrands in equation 18.

(G0,S(rB , rA) = G0,S(rA, rB)), we obtain

G∗S(rB , rA)

=

I

r∈∂V

1

iωρ[G∗

S(r, rB)∇G0(r, rA)

+ G0(r, rA)∇G∗S(r, rB)] · dS

+

Z

r∈V

iω(κ0 − κ)G∗0(r, rB)G0(r, rA)dV

+

Z

r∈V

iω(κ0 − κ)G∗S(r, rB)G0(r, rA)dV . (20)

Equations 19 and 20 are similar in form to the expres-sions derived by Wapenaar et al. (2006) and Snieder(2007) and Snieder et al. (2007). In those studies, thecross-correlation of recorded waves leads to a super-position of causal and anticausal wavefields G or G0.For attenuative media, we derive an integral expressionanalogous to equation 19 in Appendix B. In attenuativemedia, apart from the integrals in equation 19, it is nec-essary to evaluate two other volume integrals to retrieveGS(rB , rA) (see equation B1 in Appendix B).

There are two important differences between equa-tions 19 and 20 and previous expressions for Green’sfunction retrieval (Wapenaar et al., 2006; Snieder etal., 2007). The first difference is that here we obtainthe wavefield perturbations GS, which by themselvesdo not satisfy the acoustic wave equations (e.g., equa-tion 2), from cross-correlations of GS with G0. Sec-ond, the proper manipulation of unperturbed waves G0

and perturbations GS in the integrands of equations 19and 20 allows for the separate retrieval of causal andanticausal wavefield perturbations GS(rB , rA) in the

frequency-domain rather than their superposition. Sincethe correlation-type representation theorems for G orG0 (Wapenaar et al., 2006; Snieder et al., 2007) result inthe superposition of causal and anticausal responses inthe frequency-domain, their time-domain counterpartsretrieve two-sides of the signal, i.e., they retrieve thewavefield at both positive and negative times. Becauseof this, we refer to the theorems in refs. (Wapenaar etal., 2006; Snieder et al., 2007; Snieder, 2007) as two-sided

theorems. The theorems in equations 19 and 20 recoverthe time-domain field perturbation response for eitherpositive (equation 19) or negative (equation 20) timesonly. Therefore, we call the theorems in equations 19and 20 one-sided theorems.

To address the physical meaning of the differentterms in equation 19, we break our discussion apartinto the analysis of different scenarios for medium prop-erties, acquisition geometries and boundary conditions.In the first scenario, which we refer to as Case I (Fig-ure 2), the medium perturbations (including scatterers)are restricted to a volume P which is located somewhereaway from the two observation points. In this case, equa-tion 19 becomes

GS(rB , rA)

=

I

r∈∂V

1


0(r, rB)

+ G∗0(r, rB)∇GS(r, rA)] · dS

+

Z

r∈P

iω(κ0 − κ)G0(r, rA)G∗0(r, rB)dV

+

Z

r∈P

iω(κ0 − κ)GS(r, rA)G∗0(r, rB)dV ; (21)


Figure 3. Cartoon illustrating Case II. The medium con-figuration in this case is the same as for Case I (Figure 2),but now one of the receivers (e.g., rB) is placed inside theperturbation volume P.

Figure 4. Schematic representation of Case III. In this case,the medium perturbation occupies all of the volume V. As inFigure 2, solid and dashed arrows denote unperturbed wavesand field perturbations, respectively.

since κ − κ0 = 0 outside of P. Note that the integralsin equation 21 are Fourier-type oscillatory integrals,and can be evaluated asymptotically by the stationary-phase method (Bleistein & Handelsman, 1975). Thus,the dominant contribution to the integrals come fromthe stationary points along the paths of the differentfields in the integrands of equation 21. These paths areillustrated by the arrows in Figure 2. For some sourcesr ∈ ∂V (e.g., r1 in Figure 2a), the contribution of thefirst volume integral in equation 21 is effectively zerobecause the paths of the recorded unperturbed wavesin the integrand are not affected by the perturbationsin P (i.e., κ − κ0 = 0 at and close to the paths indi-cated by solid lines). This is not true for the remaining

sources r ∈ ∂V (e.g., r2 in Figure 2b) for which theobserved unperturbed waves pass through P, where thecontributions of both volume integrals in equation 21are important. Let us now consider only the sourcesr ∈ ∂V1, where ∂V1 is a segment of ∂V that containsall source positions for which the paths of G0(r, rA,B)(equation 21) are not affected by P. As discussed abovewe can then neglect the first volume integral in equa-tion 21. The last volume integral is of higher order in themedium perturbation than the surface integral. Thus,in Case I, for r ∈ ∂V1 and to leading order in V , equa-tion 21 can be reduced to

GS(rB , rA) ≈

Z

r∈∂V1

1


0(r, rB)

+ G∗0(r, rB)∇GS(r, rA)] · dS . (22)

For this special case, equation 22 is convenient becauseit allows for the reconstruction of GS(rB , rA) by cross-correlating observed quantities due to sources over ∂V1

without prior knowledge of the perturbations in P. Thishas applications in the retrieval of the medium’s per-turbation response from coherent surface sources, as wediscuss in detail in the next Section. It is importantto point out that although the truncation of the sur-face integration domain in equation 21 results in equa-tion 22, this type of truncation can introduce artifacts inGS(rB , rA) if the medium perturbations introduce mul-tiple scattering (Snieder et al., 2006; Wapenaar, 2006).Below (along with Appendix A) we provide a stationary-phase derivation to a simple model that illustrates therole of the surface and volume integrals in Case I.

In general, the volume integrals in equation 19 can-not be ignored. Let us consider another example, Case

II, illustrated by Figure 3. Now the medium configura-tion is the same as in Case I, but one of the receiverslies inside the perturbation volume P. So for Case II, itis impossible to find source positions on ∂V for whichthe direct waves observed at rB are not affected by themedium perturbation. Therefore, all integrals in equa-tion 21 must always be evaluated. Another such exampleis Case III in Figure 4, where the perturbations occurover the entire volume V.

Another interesting case (Case IV) arises when wetake the medium configuration to be the same as in Case

I (Figure 2), except the perturbation is now character-ized by a smooth wavespeed change, i.e., there are noscattered or reflected waves. In this case GS(rB , rA) =0, and equation 21 becomes

R

r∈Piω(κ0 − κ)GS(r, rA)G∗

0(r, rB)dV

=H

r∈∂V

1


0(r, rB)] · dS

+H

r∈∂V

1

iωρ[G∗

0(r, rB)∇GS(r, rA)] · dS .

(23)

In Case IV,GS(rB , rA) = 0, and equation 23 shows thatwaves observed at rA and rB due to sources over the


surface ∂V can be used to reconstruct the perturbationsinside the volume.

The last case we consider, Case V, is when homo-geneous boundary conditions apply on ∂V so that con-tribution of the surface integral vanishes. These are thesame conditions listed in the derivation of equation 17.Without the surface integral, and for arbitrary experi-mental and medium configurations, equation 19 is

GS(rB , rA) =

Z

r∈V

1

iωρG(r, rA)V (r)G∗

0(r, rB)dV .

(24)This equation, as we discuss on the next Section, playsan important role in the retrieving GS(rB , rA) from ran-dom sources within the volume V. Note that equation 24is similar to the Born approximation, which follows byreplacing G is the right hand side of expression 17 bythe unperturbed wave:

GS(rB , rA) =

Z

r∈V

1

iωρG0(r, rA)V (r)G0(r, rB) dV .

(25)The only difference between these expressions is thecomplex-conjugation of the reference field G∗

0(r, rB)in the integrand. Just as in the classical form of theLippmann-Schwinger integral in equation 17, equa-tion 24 describes GS(rB , rA) describes scattering inter-actions between the scattering potential and the un-perturbed fields in the integrand. The scattered fieldGS(rB , rA) can be equally described by causal and an-ticausal reference fields G0(r, rB) (equation 25) andG∗

0(r, rB) (equation 24). It thus follows that in thetime domain GS(rB , rA) is equally described by eithertime-advanced (i.e., causal) or time-retarded (i.e. anti-causal) reference fields G0. This property in fact holdsfor any kind of wave propagation, and is restricted tosystems that are invariant under time-reversal (i.e., inthe absence of attenuation) as discussed in (Vasconce-los, 2007).

Finally, we turn our attention to the reciprocity the-orem in equation 14. Using the Green’s functions as de-fined in equation 16, we express equation 14 as

I

r∈∂V

[GS(r, rA)∇G∗S(r, rB)] · dS +

I

r∈∂V

[G∗S(r, rB)∇GS(r, rA)] · dS

= iω

Z

r∈V

(κ− κ0)G0(r, rA)G∗S(r, rB) dV

−iω

Z

r∈V

(κ− κ0)GS(r, rA)G∗0(r, rB) dV .

(26)

This equation relates the surface integral of cross-correlated wavefield perturbations to a volume integralwhose integrand is proportional to the medium per-turbation and cross-correlations between unperturbedwaves and wavefield perturbations. Note that equa-tion 26 does not depend on the source term q (equa-

Figure 5. Illustration of the example used in the stationary-phase derivation. The unperturbed medium is homogeneous,and perturbation volume P is described by a thin sheet (solidblack line; where κ − κ0 6= 0) that extends infinitely in thehorizontal [x, y]-plane. The boundary ∂V is chosen such thatits shape is described by a cube with round edges.

Figure 6. By extending the dimensions of ∂V in Figure 5to infinity in the x- and y-directions, the integration over ∂V

can separated into two integrals: one over the top plane ∂Vt

and another over the bottom plane ∂Vb.

tion 15); the same holds for the reciprocity theoremin equation 14. Although not immediately applicablefor the reconstruction of the Green’s function as equa-tions 19 or 20, equation 26 is suitable for other purposesin remote sensing experiments. These purposes are dis-cussed in the next Section.

Example of Case I.

We present here a stationary-phase derivationwhere we evaluate all terms in equation 21 for the modelin Figure 5. In this model, the background medium isa homogeneous space and the perturbation volume P ischaracterized by a thin-sheet where κ − κ0 6= 0 and is


(a) (b)Figure 7. Cartoons representing the stationary-phase evaluation of the integral over the top plane ∂Vt. We choose the depth ofthe integration plane to be z = 0; while the perturbation sheet lies at z = D. In panel (a) we depict an arbitrary source positionr, for which the corresponding unperturbed and scattered waves are denoted by the solid and dotted arrows, respectively. Theangle ψB is defined between the direct-wave path and the vertical at r, and θA is the incidence angle of the reflected wave. Thepoint r in panel (b) is the stationary source point on ∂Vt that yields the causal scattered wave that propagates from rB to rA.Panel (b) illustrates the stationary condition which requires ψB = θA = θ.

constant. This sheet extends infinitely in the horizontal[x, y]-plane and is placed far away from the observationpoints. The medium perturbation is such that it gener-ates scattering. Since the perturbation sheet is thin (i.e.,its thickness is orders of magnitude smaller than thedominant wavelength), waves reflected off the sheet canbe thought of as single-scattered waves. Likewise, thewaves transmitted across the sheet differ from unper-turbed direct-waves only by a transmission amplitudefactor.

To perform our stationary-phase evaluation, we be-gin with equation 21, which is of the form

GS(rB , rA) =

I

r∈∂V

[ ] · dS +

Z

r∈V

[ ] dV ; (27)

where for brevity we omit the integrands of equation 21.Although in principle it is possible to evaluate boththe surface and the volume integrals on the right-handside of equation 27 using the stationary-phase method(Bleistein & Handelsman, 1975), instead we evaluate thevolume integral according to

Z

r∈V

[ ] dV = GS(rB , rA) −

I

r∈∂V

[ ] · dS . (28)

The total surface ∂V in Figure 5 consists of two sur-faces ∂Vt and ∂Vb parallel to the x, y-plane on the topand bottom of the perturbation (Figure 6). The totalcontribution of the surface integral is given by

I

r∈∂V

[ ] · dS =

Z

r∈∂Vt

[ ] · dS +

Z

r∈∂Vb

[ ] · dS . (29)

For simplicity, in this example we consider wavesonly in the far-field regime (see Appendix A and Born

& Wolf (1959). Moreover, we simplify the surface in-tegral in equation 21 by imposing the far-field radia-tion boundary condition introduced by Wapenaar et al.(2005), where ∇G · dS = (iω/c)GdS. Under these con-ditions, the integration over the top surface (AppendixA) is

Z

r∈∂Vt

2

ρcGS(r, rA)G∗

0(r, rB) dS

= −1

iω cosθGS(rB , rA) ; (30)

where θ is the scattered-wave incidence angle at the sta-tionary point (Figure 7b). The equation above showsthat with the top sources alone we can retrieve the de-sired response GS(rB , rA). This is an example of thecase described by equation 22 and illustrated more gen-erally by Figure 2a. For all arbitrary sources on the topsurface (Figure 7a), the dominant contribution to equa-tion 30 comes from the stationary source position repre-sented in Figure 7b, where the path of the unperturbedwave observed at rB is aligned with the portion of thedown-going lag of the reflected wave recorded at rA thatpasses through rB . Note that to obtain GS(rB , rA) fromequation 30, we must multiply it by −iω cosθ. The mul-tiplication by iω translates into a time-domain differ-entiation (Snieder et al., 2006; Snieder, 2007). Multi-plying equation 30 by cosθ is possible for this theoret-ical example, but unpractical in an experiment wherethe depth of the sheet is unknown. If the sheet is farfrom the observation points such that cosθ ≈ 1, equa-tion 30 (along with the time-domain differentiation)yields GS(rB , rA) exactly. When the sheet is close tothe observation points and cosθ ≈ 0, the amplitude of


(a) (b)

Figure 8. Schematic representation of the stationary-phase evaluation of the bottom-plane (∂Vb) integral. Here, the depth ofthe bottom integration plane is z = 2D. As in Figure 7, panel (a) illustrates an arbitrary source position r which sends anunperturbed wave (solid arrow) to rB and a wave perturbation (dotted arrow) to rA. ψ′

A and ψ′B are the angles defined between

the vertical at r, and the paths represented by the dotted and solid arrows, respectively. Panel (b) shows the source positionthat satisfies the stationarity condition ψ′

A = ψ′B = ψ′.

GS(rB , rA) is severely distorted (equation 30). As wediscuss below, the proper compensation for this effectcomes from the volume integral in equations 27 or 24.

The asymptotic evaluation of the integral over thebottom surface gives

Z

r∈∂Vb

2

ρcGS(r, rA)G∗

0(r, rB) dS

= −t

iω cosψ′G∗

0(rB , rA); (31)

where t is a transmission coefficient (Appendix A), andthe angle ψ′ is the stationary-phase angle defined inFigure 8b. The bottom integration results in an arrivalcorresponding to the unperturbed wave that propagatesbetween rB and rA. Because of the complex conjuga-tion in equation 31, the resulting arrival is anticausal inthe time-domain. The bottom surface integral in equa-tion 31 results in G0 instead of the desired GS(rB , rA),so its contribution must be canceled by the volume in-tegral (e.g., equation 27). This shows that for sourceson the bottom surface, the volume integrals cannot beignored in reconstructing GS . This case is an exampleof Case II (i.e., for source positions like r2) in Figure 2b.

Once the surface integrals in equations 30 and 31are evaluated, we can derive the contribution of the vol-ume integrals using equation 28. Moreover, because weseparated the contributions from the top and bottomsurfaces, we can also separate the volume integral con-tributions that correspond to the stationary points oneach surface. For sources on the top surface (Figures 6and 7), only one of the two volume integrals in equa-tion 21 is nonzero, following the discussion regarding

Case I. So, using equations 28, 30 and 21 we obtainZ

r∈V

iω(κ0 − κ)GS(r, rA)G∗0(r, rB) dV

=

„

1 +1

iω cosθ

«

GS(rB , rA) . (32)

For sources on the bottom surface both volume integralsin equation 21 must be taken into account. Therefore,from equations 28, 31 and 21,

Z

r∈V

iω(κ0 − κ)G(r, rA)G∗0(r, rB) dV

=t

iω cosψG∗

0(rB , rA) . (33)

The volume integral in equation 32 provides the am-plitude correction necessary for the result of the surfaceintegral in equation 30 to be equal to GS(rB , rA). Whilethe integral in equation 33 cancels the undesired arrivalgenerated by the surface integral in equation 31.

4 EXTRACTING THE PERTURBED

GREEN’S FUNCTION

Retrieving GS from random sources in V

The first application we propose for the represen-tation theorems presented in the previous Section ariseswhen homogeneous boundary conditions apply at thesurface, canceling the surface integral in equation 19.This yields

GS(rB , rA) =

Z

r∈V

iω(κ0 − κ)G(r, rA)G∗0(r, rB)dV .

(34)


When these homogeneous boundary conditions apply(see derivation of equation 17), the pressure observedat any given observation point rO is given by

p0(rO) =

Z

G0(r, rO)q(r)dV

p(rO) =

Z

G(r, rO)q(r)dV ; (35)

for the unperturbed and perturbed states, respectively.q is the source term in equations 1 and 2. Next we con-sider random uncorrelated sources throughout the vol-ume V, such that

q(r1, ω)q(r2, ω) = ∆κ(r1, ω) δ(r1 − r2) |R(ω)|2 ; (36)

where ∆κ = κ0 − κ and |R(ω)|2 is the power spectrumof a random excitation function. Note from equation 36that the source intensity is proportional to the local per-turbation ∆κ at every source position. We then multiplyequation 34 by |R(ω)|2 to obtain

GS(rB , rA)|R(ω)|2

= iω

Z Z

∆κ(r1, ω) δ(r1 − r2) |R(ω)|2

G(r1, rA)G∗0(r2, rB)dV1dV2

= iω

Z

G(r1, rA)q(r1)dV1

„Z

G0(r2, rB)q(r2)dV2

«∗

. (37)

Using the definitions in equation 36, equation 37 yields

GS(rB , rA) =iω

|R(ω)|2p(rA)p∗0(rB) . (38)

This equation shows that the perturbation response be-tween rB and rA can be extracted simply by cross-correlating the perturbed pressure field observed at rA

with the unperturbed pressure measured at rB . Thiscross-correlation must be compensated for the spectrum|R(ω)|2 and multiplied by iω (i.e., differentiated withrespect to time). The result in equation 38 also holdsfor perturbed attenuative media, as we show with equa-tion B4 in Appendix B. Unlike the volume sources inthe lossless case in equation 36, the random volumesources in attenuative media (equation B3 in AppendixB) are locally proportional to both the real and imag-inary parts of the compressibility perturbation, and tothe attenuation in the unperturbed medium.

Equation 38 is not practical because volume sourceslike those described by equation 36 are not easily re-producible in either lab or field experiments. The realusefulness of equation 38 lies in understanding the en-ergy partitioning requirements for the reconstruction ofthe desired scattered-wave response. Let us consider, forexample, a medium configuration like the one describedby Figure 2. In that case, according to equation 36, thevolume sources would only exist inside the perturbationvolume P. This results in a nonzero net energy flux thatis outgoing at the boundary of P. As a consequence,

there is also a preferred direction of energy flux at theobservation points. This situation is completely differ-ent than the condition of equipartitioning required forthe reconstruction of G0 or G (Wapenaar et al., 2006;Snieder et al., 2007). which requires that the total en-ergy flux within any direction at the receivers to beequal to zero. Vasconcelos (2007) provides a detaileddiscussion on the energy considerations of retrieving themedium’s perturbation responses for general physicalsystems.

Retrieving GS from coherent sources on ∂V

Equations 22 yields GS(rB , rA): the perturbedwaves that propagate between rB and rA, as if the obser-vation point at rB acts as a source. This is accomplishedby the cross-correlation of unperturbed fields observedat rB with field perturbations at rA, excited by sourceson the surface ∂V1 (equation 22). We consider coherentsources over ∂V1 such that the observed fields are givenby p(r, rA,B) = W (r, ω)G(r, rA,B) and v(r, rA,B) =(iωρ)−1W (r, ω)∇G(r, rA,B), where W (r, ω) is an exci-tation function. The observed unperturbed fields are de-fined analogously to the perturbed ones. In this case, thedesired response GS(rB , rA) can be obtained by

GS(rB , rA) ≈

Z

r∈∂V1

F(ω) [pS(r, rA)v∗0(r, rB)

+ p∗0(r, rB)vS(r, rA)] · dS ; (39)

where F(ω) = F(r, ω) is a waveform shaping filter de-signed to suppress the effect of |W (r, ω)|2, and is ideallygiven by F(r, ω) = [iωρ|W (r, ω)|2]−1.

There are experiments where the measurement ofthe particle velocity field is not available (e.g., in mostmarine seismic exploration surveys). In that case we canuse a far-field radiation condition to set ∇G · dS =(iω/c)GdS (Wapenaar et al., 2005), which simplifiesequation 22 to

GS(rB , rA) ≈

Z

r∈∂V1

2

ρcGS(r, rA)G∗

0(r, rB) dS . (40)

Then using the same observed fields present in equa-tion 39, we can extract GS(rB , rA) by

GS(rB , rA) ≈

Z

r∈∂V1

F ′(ω) pS(r, rA)p∗0(r, rB) dS ;

(41)where now F ′(ω) = F ′(r, ω) is a shaping filter with thesame objective of the filter F(ω) in equation 39, givenby F ′(r, ω) = 2[ρc|W (r, ω)|2]−1.

5 DISCUSSION AND CONCLUSION

By manipulating the acoustic wave equations for un-perturbed and perturbed media we derive convolution-and correlation-type reciprocity theorems for perturbedacoustic media. These theorems differ from previousforms of reciprocity theorems (de Hoop, 1988; Fokkema& van den Berg, 1993) because they provide explicit


relations between the wavefields and wavefield pertur-bations. Note that the wavefield perturbations by them-selves do not satisfy the perturbed wave equations.We present a suite of integral equations based on theperturbed acoustic wave equations that may be usefulboth for theoretical considerations and for applicationsin imaging acoustic scattered waves and in inferringmedium perturbations.

When the fields are described by Green’s functionsobserved at two distinct points, one of the convolution-type theorems presented here retrieves the field per-turbations that propagate between these two pointsthrough a volume integral that is similar in form tothe Lippmann-Schwinger integral (Rodberg & Thaler,1967). This convolution-type theorem result in theLippmann-Schwinger integral when the two observationpoints coincide. With the Green’s function form of thecorrelation-type representation theorems, we show thatby cross-correlating wavefield perturbations measuredat one receiver with unperturbed waves recorded by an-other we obtain the wavefield perturbations that prop-agate between the receivers as if one of the receiverswere a source. This concept relates to previous workin the field of Green’s function retrieval from diffuse-wave correlation (Weaver & Lobkis, 2001; Malcolm etal., 2004; Larose et al., 2006) and from correlation of de-terministic wavefields (Wapenaar, 2004; Snieder, 2004;Wapenaar & Fokkema, 2006). These studies show thatcross-correlations can be used to recover a superpositionof the causal and anticausal parts of the wavefields Gor G0 (i.e., unperturbed or perturbed). Our expressionsrecover the wavefield perturbations GS separately fromits anticausal counterpart G∗

S. For systems that are in-variant under time reversal, Green’s function retrievalby wavefield cross-correlations require only a surface in-tegration (Larose et al., 2006; Wapenaar & Fokkema,2006; Snieder et al., 2007), whereas the retrieval of theperturbations GS from correlations of wavefield pertur-bations with unperturbed wavefields requires an addi-tional volume integral.

The extraction of wavefield perturbations GS thatpropagate between receivers as if one of them acts asa source can be a useful tool for remote sensing exper-iments such as those in medical imaging, ocean acous-tics and seismology, for example. When medium per-turbations are localized within a volume P that doesnot contain the observation points, the wavefield per-turbations propagating between receivers can be ob-tained only from the surface integral of cross-correlatedperturbations and unperturbed wavefields (i.e., the vol-ume integral can be neglected). Neglecting the volumeintegrals can yield distortions in the reconstructed re-sponse. The extracted perturbed Green’s function canbe used for the purpose of detecting, locating or moni-toring medium perturbations.

When the boundary conditions are such that thesurface integral vanishes, the field perturbations propa-

gating between two receivers can be extracted from thecross-correlation of unperturbed fluctuations measuredat one receivers with the perturbed response measuredat the other. This only holds if the measured responsesare due to sources distributed throughout the entire vol-ume that are locally proportional to the medium per-turbation. Although this result is of limited practicalapplications, it gives an important insight into the en-ergy considerations for reconstructing only the field per-turbations. While energy equipartitioning is requiredfor the accurate reconstruction of the perturbed or un-perturbed wavefields, the reconstruction of the wave-field perturbations (e.g., scattered waves only) requiresa non-zero net energy flux at the observation points.This is discussed in detail in Vasconcelos (2007).

We have proposed direct applications of the inter-ferometric retrieval of wavefield perturbations as pro-posed in this paper (Vasconcelos & Snieder, 2007; Vas-concelos et al., 2007). Vasconcelos et al. (2007) use theretrieval of wavefield perturbations according to theformulation in this paper to image salt structures inthe Earth’s subsurface from the interference of multi-ply scattered waves measured inside a deep boreholein off-shore Gulf of Mexico. Vasconcelos and Snieder(2007) use the concepts developed here to interpret thephysical meaning of their deconvolution-based interfer-ometry approach. Their deconvolution interferometrymethod (based partly on equations 21 and 40) has beenvalidated with numerical experiments (Vasconcelos &Snieder, 2008; Vasconcelos et al., 2007a, Vasconcelos etal., 2007b) and has been applied to imaging with drill-bit noise.

The interferometric retrieval of wavefield perturba-tions we describe here can be used for targeting theimaging of particular portions of the medium (Vas-concelos et al., 2007a) where this technique is calledtarget-oriented interferometry. Such an application hasalso been implicitly proposed by Bakulin and Calvert(2006) and by Mehta et al. (2007), in the so-calledVirtual Source method. With this method, transmis-sion and reflection responses can be regarded as unper-turbed waves and wavefield perturbations, respectively.Although most of the examples cited come from the fieldof geophysics, our results are immediately applicable toother fields in acoustics such as physical oceanography,laboratory and medical ultrasonics, and non-destructivetesting.

Another important potential use for the exact formof the correlation-type representation theorems that ex-press the wavefield perturbations that propagate be-tween two receivers lies in the calculation of Frechet

derivatives (Tarantola, 1987; Hettlich, 1998; Sava &Biondi, 2004), which consist of the partial derivatives ofthe wavefield perturbations with respect to the mediumperturbations. These derivatives which can be derivedfrom the theorems we provide here. For example, in-troducing an infitesimal medium parameter change in


equation 25 yields the derivative ∂GS/∂κ(r). Thesederivatives are important for the computation of sen-sitivity kernels used in full-waveform inversion (Taran-tola, 1987), in imaging (Hettlich, 1998) or in linearizedformulations of wave-equation based tomography (Sava& Biondi, 2004).

From correlation- and convolution-type representa-tion theorems in perturbed media, we suggest the appli-cation of estimating the medium perturbations by com-bining theorems presented here with inverse scatteringtheory (Weglein, et al., 2003; Malcolm et al., 2004).This type of approach can potentially be used for imag-ing of the wavefield perturbations (Tarantola, 1987; We-gleian, et al., 2003), as well as for the targeting the ex-traction of a particular desired subset of the wavefieldperturbations (Malcolm et al., 2004). In this context,the use of our expressions can be simplified by lineariz-ing the wavefield perturbations in the medium changes(one such approximation is given by equation 25). Thiswould yield, for example, a Born approximation (Born& Wolf, 1959; Snieder, 1990; Weglein et al., 2003) of thetheorems presented here.

Apart from imaging applications, our results (bothin terms of retrieving wavefield perturbations and forestimating medium perturbations) can be used for mon-itoring temporal changes in the medium. In geoscience,this could be applied to remotely monitoring the deple-tion of aquifers or hydrocarbon reservoirs; or monitoringthe injection of CO2 for carbon sequestration. In mate-rial science, our results can be used to monitor materialintegrity with respect, for example, to temporal changesin temperature or changes due to crack formation. Thedetection of earthquake damage is a potential applica-tion in the field of structural engineering. Within med-ical imaging applications, our expressions can be tai-lored, for instance, to observe tumor evolution from aseries of time-lapse ultrasonic measurements.

6 ACKNOWLEDGMENTS

This research was financed by the NSF (grant EAS-0609595) and by the sponsors of the CWP ConsortiumProject. We are thankful to Huub Douma (PrincetonUniversity) for his detailed review of this manuscript.Huub helped us clarify the complimentary aspects ofour research.

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APPENDIX A: STATIONARY-PHASE

EVALUATION OF THE THIN-LAYER CASE.

Integration over ∂Vt.

The wavefields shown in Figure 7 can be describedby the ray-geometric impulse responses

G0(r, rB) = −ρeik|rB − r|

4π |rB − r|, and

GS(r, rA) = −rρeik(|rsr

A− rA|+ |r− r

sr

A |)

4π (|rsrA − rA| + |r − rsr

A |);(A1)

where rsrA are the specular reflection points for the

receiver-source pairs (r, rA). G0 and GS are the far-fieldacoustic Green’s functions we use to describe direct- andspecularly-reflected waves, respectively. r is the reflec-tion coefficient, which for simplicity we assume to beconstant with respect to incident angle. In our model,r = (x, y, z = 0) (for r ∈ ∂Vt) and rA,B = (xA,B, yA,B =0, zA,B). The distances in the phases and denominatorsin equation A1 can be expressed in terms of the cor-responding ray-lengths in Figure 7. Using the Green’sfunctions in equation A1 to express

I∂Vt=

Z

r∈∂Vt

2

ρcGS(r, rA)G∗

0(r, rB) dS , (A2)

we get

I∂Vt=

2ρr

c (4π)2

Z

eik(L1+L2−LB)

(L1 + L2)LBdxdy , (A3)

where ϕ = ik (L1 + L2 − LB) is the phase of the inte-grand, and k = ω/c. LB = |rB − r| is the direct-wavepath in Figure 7; and L1 = |rsr

A − r| and L2 = |rA − rsrA |

are, the down- and up-going lags of the reflected-wavepath in Figure 7, respectively. The source position that

gives a stationary contribution to the integral in equa-tion A3 satisfies (Snieder et al., 2006)

θA = ψB (A4)

and

y = 0, (A5)

where ψB is the angle defined between the direct waveand the vertical at rB , and θA = θ is the incidenceangle, at the interface, of the reflected wave recorded atrA (Figure 7).

Because yA,B = 0 and the model is a flat reflectorin a homogeneous and isotropic medium, the station-ary contribution for all terms comes from sources aty = 0 .The condition θA = ψB (equation A4) statesthat the stationary source is the one that sends a directwave which is passes straight through rB , gets reflectedand is then recorded at rA. This is the same stationarycondition as for the C2

AB term analyzed by Snieder etal.(2006).

To approximate the integral in equation A3 withthe stationary-phase method we must evaluate, at thestationary point, the second derivatives

∂2L1

∂x2=

L1

(L1 + L2)2 cos

2θ , (A6)

∂2L2

∂x2=

L2

(L1 + L2)2 cos

2θ , (A7)

∂2LB

∂x2=

1

LBcos2θ , (A8)

and

∂2L1

∂y2=

L1

(L1 + L2)2 , (A9)

∂2L2

∂y2=

L2

(L1 + L2)2 , (A10)

Based on these second derivatives, the zero-orderstationary-phase approximation (Bleistein and Handels-man, 1975) to equation A3 is

I∂Vt=

2ρr

c (4π)2eik(L1+L2−LB)

(L1 + L2)LB

“

e−iπ/4”2 2π

k cosθ

×

„

1

LB−

1

L1 + L2

«−1

;

(A11)which yields

I∂Vt=

ρr

−iω cosθ (4π)

eik(L1+L2−LB)

(L1 + L2 − LB). (A12)

At the stationary source point, where θA = ψB , the dis-tance L1 + L2 − LB is equivalent to the ray-length ofthe scattered wave that propagates between rA and rB


(see Figure 7b). Thus, in the stationary-phase approxi-mation, I∂Vt

is given by

I∂Vt=

1

−iω cosθGS(rB , rA) ; (A13)

showing how the I∂Vtterm results in a causal scattered

wave propagating from rB to rA.Integration over ∂Vb.

The stationary-phase evaluation of the bottom sur-face integral is analogous to that of the top surface in-tegral, but with the impulse responses given by

G0(r, rB) = −ρeik|rB − r|

4π |rB − r|, and

GS(r, rA) = −tρeik|rA − r|

4π |rA − r|; (A14)

where t is a transmission coefficient, such that the totaltransmission coefficient in G(r, rA) is given by (1 + t).The bottom surface integral can be then written as

I∂Vb=

Z

r∈∂Vb

2

ρcGS(r, rA)G∗

0(r, rB) dS

=2ρt

c (4π)2

Z

eik(LA−LB)

LALBdxdy , (A15)

with LB and LA being the ray-lengths corresponding tothe solid and dashed arrows in Figure 8, respectively.These lengths are defined by

LA,B =

q

(x− xA,B)2 + y2 + (2D − zA,B)2 . (A16)

Given that the phase of the integrand on the right-handside of equation A15 is controlled by ϕ′ = (LA − LB),the stationary source point can be found by setting

0 =∂ϕ′

∂y=

y

LA−

y

LB, (A17)

and

0 =∂ϕ′

∂x=x− xA

LA−x− xB

LB= sinψ′

A−sinψ′B ; (A18)

the angles ψ′A and ψ′

B are defined in Figure 8. It fol-lows from equations A17 and A18 that the phase of theintegrand of equation A15 is stationary when

ψ′A = ψ′

B = ψ′ and y = 0 . (A19)

Since the sources are restricted to the bottom surface,the condition in equation A19 is satisfied only when theray-paths in Figure 8a are aligned as in Figure 8b.

Next we evaluate, at the stationary point (equa-tion A19), the second derivatives

∂2ϕ′

∂x2=

(2D − zA)2

L3A

−(2D − zB)2

L3B

= cos 2ψ′

„

1

LA−

1

LB

«

, (A20)

and

∂2ϕ′

∂y2=L2

A

L3A

−L2

B

L3B

=1

LA−

1

LB. (A21)

The second derivatives are used to approximate the in-tegral in equation A15, according to

I∂Vb=

2ρt

c (4π)2exp (ik (LA − LB))

LALB

× e−iπ/4

r

2π

k

1s

cos 2ψ

„

1

LA−

1

LB

«

× e−iπ/4

r

2π

k

1r

1

LA−

1

LB

, (A22)

which gives

I∂Vb=

ρt

−iω cosψ′ (4π)

exp (ik (LA − LB))

(LA − LB). (A23)

Note that at the stationary point shown in Figure 8b,LB −LA is the ray-length of an unperturbed wave prop-agating between rB and rA, therefore, equation A23 canbe represented by

I∂Vb=

t

−iω cosψ′G∗

0(rB , rA) . (A24)

APPENDIX B: EXTENSION TO

ATTENUATIVE MEDIA

While in the main text we only treat the case of acousticwaves in loss-less media, incorporating attenuation intoour expression can be done in a straightforward mannerby making κ0 and κ a complex quantity (Snieder, 2007;Douma, 2008), and by replacing (κ0 − κ) by (κ∗

0 − κ)where appropriate (with the exception of equation 20,where it should be replaced by (κ0 − κ∗)). Then, forexample, equation 19 in attenuative media reads

GS(rB , rA) =

I

r∈∂V

1


0(r, rB)

+G∗0(r, rB)∇GS(r, rA)] · dS

−

Z

r∈V

iωRe˘

∆κ′¯

G(r, rA)G∗0(r, rB)dV

+

Z

r∈V

2ω Im {κ0} G(r, rA)G∗0(r, rB)dV

−

Z

r∈V

ω Im˘

∆κ′¯ G(r, rA)G∗0(r, rB)dV ;

(B1)

where Re and Im stand for the real and imaginary partsof a complex quantity, respectively, and ∆κ′ = κ∗

0 − κ.The first volume integral in equation B1 yields the in-tegrals in equation 19, while the other volume integralaccount for attenuation. Note that in attenuative me-dia, even if there’s no perturbation (i.e., ∆κ′ = 0), thesecond volume integral in equation B1 is nonzero. Thiscase is analyzed by Snieder (2007).

If homogeneous boundary conditions apply on ∂V


that cancel the contribution of the volume integral(equation 34), equation B1 simplifies to

GS(rB , rA)

=

Z

r∈V

iωRe {(κ0 − κ)} G(r, rA)G∗0(r, rB)dV

+

Z

r∈V

ω Im {(κ0 − κ)} G(r, rA)G∗0(r, rB)dV .(B2)

We now consider random volume sources similar toequation 36 but now described by

q(r1, ω)q(r2, ω) = ∆κ′(r1, ω) δ(r1 − r2) |R(ω)|2 . (B3)

Note that at every point in the volume, equation B3describes a superposition of three sources which are lo-cally proportional to i) Re {∆κ′}, ii) Im {∆κ′} and iii)Im {κ0}, respectively. Through a derivation analogousto equation 37, equation B2 gives

GS(rB , rA) =iω

|R(ω)|2p(rA)p∗0(rB) , (B4)

same as in equation 38.When the surface integral is nonzero, the volume

integrals cannot be ignored. In particular, for attenua-tive media the discussion that leads equation 22 is notvalid because the second volume integral in equation B1above in general cannot be neglected. Nevertheless, re-sults from field exploration seismology surveys (Bakulin& Calvert, 2006; Mehta et al., 2007; Vasconcelos et al.,2007) suggest that only evaluating the surface integralretrieves the desired with correct kinematics and geo-metrical spreading. It possible that in some cases (likeCase I, Figure 2) the volume integrals provide an ampli-tude correction to what is extracted from the surface in-tegral alone, but this issue is yet to be resolved (Snieder,2007).

representation theorems in perturbed acoustic media ...cwp-584p representation theorems in perturbed...

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