PHYSICAL REVIEW E 80, 036605 !2009"

Representation theorems and Green’s function retrieval for scattering in acoustic media

1 2 1

Ivan Vasconcelos, Roel Snieder, and Huub Douma

1

ION Geophysical, GXT Imaging Solutions, 1st Floor, Integra House, Vicarage Road, Egham, Surrey TW20 9JZ, United Kingdom

2

Center for Wave Phenomena, Department of Geophysics, Colorado School of Mines, Golden, Colorado 80401, USA

!Received 3 January 2009; revised manuscript received 6 July 2009; published 22 September 2009"

Reciprocity theorems for perturbed acoustic media are provided in the form of convolution- and correlation-

type theorems. These reciprocity relations are particularly useful in the general treatment of both forward and

inverse-scattering problems. Using Green’s functions to describe perturbed and unperturbed waves in two

distinct wave states, representation theorems for scattered waves are derived from the reciprocity relations.

While the convolution-type theorems can be manipulated to obtain scattering integrals that are analogous to the

Lippmann-Schwinger equation, the correlation-type theorems can be used to retrieve the scattering response of

the medium by cross correlations. Unlike previous formulations of Green’s function retrieval, the extraction of

scattered-wave responses by cross correlations does not require energy equipartitioning. Allowing for uneven

energy radiation brings experimental advantages to the retrieval of ﬁelds scattered by remote lossless and/or

attenuative scatterers. These concepts are illustrated with a number of examples, including analytic solutions to

a one-dimensional scattering problem, and a numerical example in the context of seismic waves recorded on

the ocean bottom.

DOI: 10.1103/PhysRevE.80.036605 PACS number!s": 43.20.!g, 43.40.!s, 43.60.!d, 43.35.!d

I. INTRODUCTION

from the autocorrelation of recorded transmission responses.

This result was later extended for cross correlations in het-

Reciprocity theorems have long been used to describe im-

erogeneous three-dimensional media by Wapenaar et al. #13$,

portant properties of wave propagation phenomena. Rayleigh

who used one-way reciprocity theorems in their derivations.

#1$ used a local form of an acoustic reciprocity theorem to

Green’s function retrieval by cross correlations has found

demonstrate source-receiver reciprocity. Time-domain reci-

applications in the ﬁelds of global #10,14$ and exploration

procity theorems were later generalized to relate two wave

seismology #15,16$, ultrasonics #17,18$, helioseismology

states with different ﬁeld, material, and source properties in

#19$, structural engineering #20,21$, and ocean acoustics

absorbing heterogeneous media #2$.

#22,23$.

Fokkema and van den Berg #3$ showed that acoustic reci-

Although the ability to reconstruct the Green’s function

procity theorems can be used for modeling wave propaga-

between two observation points via cross correlations has

tion, for boundary and domain imaging, and for estimation of

been shown for special cases by methods other than repre-

the medium properties. In the ﬁeld of exploration seismol-

sentation theorems !e.g., #8,16,24$", the derivations based on

ogy, an important application of convolution-type reciprocity

representation theorems have provided for generalizations

theorems lies in removing multiple reﬂections, also called

beyond lossless acoustic wave propagation to elastic wave

multiples, caused by the Earth’s free surface #3,4$. These

propagation and diffusion. More general forms of reciprocity

approaches rely on the convolution of single-scattered waves

relations have been derived #7,25,26$ which include a wide

to create multiples, which are then adaptively subtracted

range of differential equations such as the acoustic, elastody-

from the recorded data. Other approaches for the elimination

namic, and electromagnetic wave equations, as well as the

of multiples from seismic data rely on inverse-scattering

diffusion, advection, and Schrödinger equations, among oth-

methods #5$. The inverse-scattering-based methodologies are

ers.

1

typically used separately from the representation theorem

In this paper, we derive reciprocity theorems for acoustic

approaches #3,4$ in predicting multiples.

perturbed media. The perturbations of the wave ﬁeld due to

Recent forms of reciprocity theorems have been derived

the perturbation of the medium can be used for imaging or

for the extraction of Green’s functions #6,7$, showing that the

for monitoring. For imaging, the unperturbed medium is as-

cross correlations of waves recorded by two receivers can be

sumed to be so smooth that it does not generate reﬂected

used to obtain the waves that propagate between these re-

waves, while discontinuities in the perturbation account for

ceivers as if one of them behaves as a source. These results

scattering. In monitoring applications, the perturbation con-

coincide with other studies based on cross correlations of

sists of the time-lapse changes in the medium. Although pre-

diffuse waves in a medium with an irregular boundary #8$,

vious derivations of reciprocity theorems account for arbi-

caused by randomly distributed uncorrelated sources #9,10$,

trary medium parameters that are different between two

or present in the coda of the recorded signals #11$. An early

wave states #2,3,7$, they do not explicitly consider the spe-

analysis by Claerbout #12$ shows that the reﬂection response

cial case of perturbed media or scattering. In perturbed me-

in a one-dimensional !1D" medium can be reconstructed

dia, there are special relations between the unperturbed and

perturbed wave states !e.g., in terms of the physical excita-

1

tion" that make the reciprocity theorems in such media differ

Representation theorems are derived from reciprocity theorems

using Green’s functions; e.g., see Sec. III of this paper. in form with respect to their more general counterparts #3,7$.

1539-3755/2009/80!3"/036605!14" 036605-1 ©2009 The American Physical SocietyVASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

Here we focus on deriving and discussing some of these

differences.

One particularly important aspect of studying scattering-

based reciprocity lies in retrieving wave ﬁeld perturbations

from cross correlations #7,25$. As we show here, wave ﬁeld

perturbations by themselves do not satisfy the wave equa-

tions and thus their retrieval does not follow directly from

earlier derivations. More importantly, here we demonstrate

that the accurate retrieval of scattered waves by correlation

does not require energy equipartitioning as does the retrieval

of full-ﬁeld responses #7,24,25$. This is an important result

for dealing with certain remote sensing and imaging experi-

ments where only a ﬁnite aperture of physical sources is

available. Moreover, we show that this result holds both for

FIG. 1. Illustration of the domain used in the reciprocity theo-

lossless and attenuative scattering problems.

rems. The domain consists of a volume V, bounded by!V. The unit

We ﬁrst outline general forms of convolution- and

vector normal to !V is represented by n. The wave states A and B

correlation-type reciprocity theorems by manipulating the are represented by receivers placed at r !white triangle" and r

A B

perturbed and unperturbed wave equations for two wave !gray triangle", respectively. The solid arrows denote the stationary

paths of unperturbed waves G , propagating between the receivers

states. Then, we write the more general reciprocity relations

0

and an arbitrary point r on !V.

as representation theorems using the Green’s functions for

unperturbed and perturbed waves in the two states. We show

T

that the convolution-type theorem results in a familiar scat-

tion is such that "=!!/!r , ... ,!/!r " and "·v

1 d

d

tering integral that describes ﬁeld perturbations between two

=& ! /!r . The unperturbed wave equations are obtained

v

i=1 i i

observation points. Next we analyze how the correlation-

by adding the subscript 0 to coefﬁcients and ﬁeld quantities

type theorems can be used to extract the ﬁeld perturbations

in Eq. !1".

from cross correlations of observed ﬁelds for different types

We assume that no volume forces are present by setting

of media and experimental conﬁgurations. Finally, we dis-

the right-hand side !RHS" of the vector relation in Eq. !1"

cuss the applications of these representation theorems in re-

equals zero. For brevity, we assume that perturbations only

covering the perturbation response between two sensors from

occur in compressibility, thus#=# , but our derivations can

0

random medium ﬂuctuations and from coherent surface

be generalized to include density perturbation as well. We

sources. Our results are illustrated by one-dimensional ana-

make no restrictions on the smoothness of the material pa-

lytic examples and by a numerical example of the application

rameters, i.e., rapid lateral changes and discontinuities are

of scattering reciprocity to acoustic waves recorded at the

allowed.

ocean bottom.

To derive Rayleigh’s reciprocity theorem #1–3$, we insert

the equations of motion and stress-strain relations for states

II. RECIPROCITY THEOREMS IN CONVOLUTION AND A and B in

CORRELATION FORM

B A A B A B B A

d

We deﬁne acoustic wave states in a domain V!R , v · E + p E − v · E − p E , !2"

0 0 0 0 0 0 0 0

d

bounded by !V!R !Fig. 1". The outward pointing normal

to !V is represented by n. We consider two wave states,

where E and E represent, from Eq. !1", the equation of mo-

which we denote by the superscripts A and B, respectively.

tion !ﬁrst line of the equation" and the stress-strain relation

Each wave state is deﬁned in an unperturbed medium with

!second line of the equation", respectively. For brevity, we

compressibility " !r" and density # !r", as well as in a per-

0 0

omit the parameter dependence on r and$. From Eq. !2" we

turbed medium described by"!r" and#!r". Using the Fourier

A B B A

isolate the interaction quantity"·!p v −p v " #2$. Next, we

0 0 0 0

convention u!t"=%u!$"exp!−i$t"d$, the ﬁeld equations for

integrate the result of Eq. !2" over the domain V and apply

state A in a perturbed medium are, in the frequency domain,

Gauss’ divergence theorem. This results in

A A

"p !r,$"− i$#!r"v !r,$" = 0,

A A A

A B B A A B B A

" · v !r,$"− i$"!r"p !r,$" = q !r,$", !1"

#p v − p v $ · dS = #p q − p q $dV, !3"

(

’

0 0 0 0 0 0 0 0

r!!V r!V

A A

where p !r,$" and v !r,$" represent pressure and particle

d

velocity, respectively, observed at the point r!R for a

given time-harmonic frequency $!R. The perturbed ﬁelds which is referred to as a reciprocity theorem of the convolu-

for any wave state are p=p +p and v=v +v , where the

tion type #2,3$ because the frequency-domain products of

0 S 0 S

subscript S indicates the wave ﬁeld perturbation caused by ﬁeld parameters represent convolutions in the time domain.

A

medium changes. The quantity q !r,$" describes the source

A correlation-type reciprocity theorem #2,3$ can be derived

A B! B! A

distribution as a volume injection rate density and is the from isolating the interaction quantity "·!p v +p v "

0 0 0 0

same for both perturbed and unperturbed waves. Our nota-

from

036605-2REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"

B! A A B! A B! B! A

By interchanging the superscripts in Eqs. !6" and !8" we

v · E + p E + v · E + p E , !4"

0 0 0 0 0 0 0 0

derive convolution- and correlation-type reciprocity theo-

!

B B A A

where superscript denotes complex conjugation. Subse-

rems that relate the perturbations p and v to p and v .

S S 0 0

quent volume integration and application of the divergence

These theorems have the same form as the ones in Eqs. !7"

theorem yield

and !9", except A is interchanged with B in Eq. !7" and with

B! in Eq. !9". Although Eqs. !7" and !8" account for com-

A B! B! A A B! B! A

pressibility changes only, they can be modiﬁed to include

#p v + p v $ · dS = #p q + p q $dV,

’ (

0 0 0 0 0 0 0 0

r!!V r!V density perturbations. Such modiﬁcation involves adding, to

the RHS of the equations, an extra volume integral whose

!5"

A

integrand is proportional to !# −#" and the wave ﬁelds v

0

B B!

where complex conjugates translate into time-domain cross

and v !or v " #3$.

0 0

correlations of ﬁeld parameters. For this reason, Eq. !5" is a

reciprocity theorem of the correlation type #2,3$.

Convolution- and correlation-type reciprocity theorems for

III. SCATTERING-BASED REPRESENTATIONS AND

the perturbed wave states #e.g., Eq. !1"$ can be expressed APPLICATIONS

simply by removing the subscript 0 from Eqs. !2"–!5". In Eq.

We introduce the Green’s functions, in the frequency do-

!5" we assume that " and # are real quantities !i.e., the

0 0

main, by setting

medium is lossless".

The theorems in Eqs. !3" and !5" hold when the material

properties in states A and B are the same. General reciprocity

A,B d

q =%!r− r ", r !R . !10"

A,B A,B

theorems that account for arbitrarily different source and ma-

terial properties between two wave states have been derived

This choice for q allows for expressing the ﬁeld quantity p in

in #2,3$. Here, we further develop these reciprocity theorems

terms of the Green’s functions G, i.e.,

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

A B B A

isolate"·!p v −p v " from

0 0

A,B

B A A B A B B A p !r,$" = G!r ,r,$" = G !r ,r,$" + G !r ,r,$".

A,B 0 A,B S A,B

v · E + p E − v · E − p E . !6"

0 0 0 0

!11"

After separating this quantity, we integrate over r!V and

apply the divergence theorem. Using p=p +p and v=v

0 S 0

Note that these are the Green’s functions for sources of the

+v and subtracting Eq. !3", we obtain

S

volume injection rate type. The derivation below can also be

reproduced using volume forces #6$. It follows from Eqs.

A B B A A B

#p v − p v $ · dS = p q dV A,B −1

’ (

S 0 0 S S 0

!11" and !1" that v !r,$"=!i$#" "G!r ,r,$".

A,B

r!!V r!V

Using these deﬁnitions, the convolution-type theorem in

Eq. !7" becomes

A B

+ i$!" −""p p dV,

( 0

0

r!V

!7"

G !r ,r " = G !r ,r"%!r− r "dV

S A B ( S A B

r!V

which is a convolution-type reciprocity theorem for per-

turbed media.

1

= #G !r ,r"" G !r ,r"

’

S A 0 B

The correlation-type counterpart of Eq. !7" can be derived

i$#

r!!V

B! B!

A A

from the interaction quantity "·!p v +p v ", which can

0 0

be isolated from − G !r ,r"" G !r ,r"$ · dS

0 B S A

B! A A B! A B! B! A

1

v · E + p E + v · E + p E . !8"

0 0 0 0

+ G!r ,r"V!r"G !r ,r"dV, !12"

(

A 0 B

i$#

r!V

After performing the same steps as in the derivation of Eq.

2

!7" we obtain

where V!r"=#$ #"!r"−" !r"$ is the perturbation operator

0

or scattering potential #27$. For brevity we omit the depen-

A B! B! A A B!

dence on the frequency $. Now we consider this equation

#p v + p v $ · dS = p q dV

’ (

S 0 0 S S 0

under homogeneous boundary conditions on !V, namely, !i"

r!!V r!V

Sommerfeld radiation conditions #6$, !ii" Dirichlet boundary

A B!

conditions, i.e., G !r,r "=0 ∀r!!V, and/or !iii" Neu-

0,S A,B

− i$!" −""p p dV,

(

0

0

mann boundary conditions, "G !r,r "·n=0 ∀r!!V.

r!V

0,S A,B

This gives

!9"

which is a correlation-type reciprocity theorem for perturbed

1

acoustic media. Again, we assume that both" and" are real

0 G !r ,r " = G!r ,r"V!r"G !r ,r"dV. !13"

(

S A B A 0 B

i$#

r!V

!i.e., no attenuation".

036605-3VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

FIG. 2. Schematic illustrations of conﬁgurations for case I. Medium perturbations are restricted to the subdomainP, which is placed away

from the observation points. By inﬁnitely extending the sides of !V, the closed surface integral can be replaced by an integral over r

!!V #!V , as portrayed in panels !a" and !b". In our discussion, we ﬁx the sets !V and P and have two choices for !V such that in !b"

b t t b

P!V and in !c" P$V.

Equation !13" is the integral equation known as the taking either time-advanced #i.e., G !r ,r"; Eq. !13"$ or

0 B

!

Lippmann-Schwinger equation #27$, commonly used for

time-reversed #i.e., G !r ,r"; Eq. !15"$ ﬁelds.

B

0

modeling and inversion or imaging in scattering problems.

The left-hand side of Eq. !14" describes causal wave ﬁeld

When none of the surface boundary conditions listed above

perturbations that propagate from r to r as if the observa-

B A

apply, the surface integral of Eq. !7" should be added to the

tion point at r acts as a source. By taking Eq. !14", inter-

B

right-hand side of Eq. !13".

changing subscripts A by B!, and taking the complex conju-

Next, we turn our attention to the correlation-type reci-

gate, we obtain

procity theorem in Eq. !9". Substituting the Green’s functions

#Eq. !11"$ for the wave ﬁelds p and v in Eq. !9" gives

1

! !

G !r ,r " = #G !r ,r"" G !r ,r"

A B ’ B 0 A

S S

i$#

r!!V

G !r ,r " = G !r ,r"%!r− r "dV

(

S A B S A B

!

r!V − G !r ,r"" G !r ,r"$ · dS

0 A S B

1

1

!

!

= #G !r ,r"" G !r ,r"

− G !r ,r"V!r"G !r ,r"dV. !16"

’ B S A

0 ( B 0 A

i$#

i$#

r!!V

r!V

!

− G !r ,r"" G !r ,r"$ · dS

S A B There are two important differences between Eqs. !14"

0

and !16" and previous expressions for Green’s function re-

1

!

+ G!r ,r"V!r"G !r ,r"dV. !14" trieval #7,25$. The ﬁrst difference is that here we obtain the

(

A 0 B

i$#

r!V

wave ﬁeld perturbations G , which by themselves do not

S

satisfy the acoustic wave equations #e.g., Eq. !1"$, from cross

correlations of G with G . Second, the proper manipulation

S 0

of unperturbed waves G and perturbations G in the inte-

0 S

The surface integral here does not vanish under a Sommer-

grands of Eqs. !14" and !16" allows for the separate retrieval

feld radiation condition, but with Dirichlet and/or Neumann

of causal and anticausal wave ﬁeld perturbations G !r ,r "

S A B

boundary conditions we get

in the frequency domain rather than their superposition.

Since the correlation-type representation theorems for G or

G #7,25$ result in the superposition of causal and anticausal

0

1 responses in the frequency domain, their time-domain coun-

!

G !r ,r " = G!r ,r"V!r"G !r ,r"dV, !15"

(

S A B A 0 B

terparts retrieve two sides of the signal, i.e., they retrieve the

i$#

r!V

wave ﬁeld at both positive and negative times. Because of

this, we refer to the theorems in Refs. #7,25$ as two-sided

theorems. The theorems in Eqs. !14" and !16" recover the

time-domain ﬁeld perturbation response for either positive

which is similar to the Lippmann-Schwinger integral in Eq.

!13", except for the complex conjugate in the RHS. Under #Eq. !14"$ or negative #Eq. !16"$ times only. Therefore, we

Neumann and/or Dirichlet boundary conditions, inspection call the theorems in Eqs. !14" and !16" one-sided theorems.

of Eqs. !13" and !15" states that modeling and inversion or Let us consider a ﬁrst scenario, which we refer to as case

imaging of scattered ﬁelds can be accomplished equally by I !Fig. 2", deﬁned by

036605-4REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"

d

!i" V!r"" 0; only for r!P; P!R

!ii" sing supp#V!r"$" 0; !i.e., V generates backscattering"

!iii" r "P; !i.e., perturbations away from receiver acting as source"

B

!17"

& d

!iv" " !r", # !r"! C !R ";

!i.e., smooth background",

0 0

−1

!v" #iG !r,r "$ " G !r,r " · n!r "’ 0; for !r,r "!!V or !V !i.e., outgoing reference waves"

0 s 0 s s s b t

−1

#iG !r,r "$ " G !r,r " · n!r "( 0; for !r,r "!!V !i.e., ingoing scattered waves".

S s S s s s b

In this case, Eq. !14" becomes

1

! !

#G !r ,r"" G !r ,r"− G !r ,r"" G !r ,r"$ · dS

( B S A S A B

0 0

i$#

r!!V

b

1

G !r ,r " =

S A B (

1

i$# !

r!!!V #!V "

=− G!r ,r"V!r"G !r ,r"dV. !22"

b t

( A B

0

i$#

r!V

!

)#G !r ,r"" G !r ,r"

B S A

0

In case I #Figs. 2!b" and 2!c"$, it follows from Eq. !21"

!

− G !r ,r"" G !r ,r"$ · dS

S A B

0

that we can retrieve the exact scattered ﬁeld G !r ,r " be-

S A B

tween two sensors by cross-correlating reference and scat-

1

!

+ G!r ,r"V!r"G !r ,r"dV, !18"

tered waves only from sources on the open top surface !V .

( A B

0 t

i$#

r!P

Moreover, Eqs. !19", !20", and !22" demonstrate that the vol-

ume integral in Eq. !18" exists only to account for medium

assuming that P!V #Fig. 2!b"$. Note here that the integra-

perturbations that lie between surface sources and the re-

tion is now carried out for sources on the open top surface

ceiver that acts as a pseudosource !i.e., r in this case".

B

!V and on the bottom surface !V !Fig. 2". If P$V #Fig.

Therefore, in any practical conﬁguration of case I, the

t b

2!c"$, then V!r"=0 ∀r!V, which results in

bottom-surface sources and the volume integral can simply

be neglected. This also implies that the observation points r

A

could be anywhere !even inside P". We illustrate how this

1

!

G!r ,r"V!r"G !r ,r"dV = 0. !19"

(

A 0 B

observation holds for different source-receiver conﬁgurations

i$#

r!V

with one-dimensional analytic examples !see below".

In general, the volume integrals in Eq. !14" cannot be

Furthermore, if P$V as in Fig. 2!c",

ignored. Let us consider another example, case II, illustrated

! −1 ! ! −1

#iG !r,r "$ "G !r ,r"·n(0 and #iG !r,r "$ "G !r ,r"

s B s S A

0 0 S

by Fig. 3. The conﬁguration is similar to that of case I #see

·n’0 for all r!!V #see conditions in Appendix A and Eq.

b

conditions in Eq. !17"$, but now condition !iii" in Eq. !17" is

!17"$, giving

modiﬁed to r !P. So for case II, it is impossible to ﬁnd

B

source positions on !V for which there are waves whose

1

! !

#G !r ,r"" G !r ,r"− G !r ,r"" G !r ,r"$ · dS

( B S A S A B

0 0

i$#

r!!V

b

= 0, !20"

because the effective contributions of the two integral terms

cancel !i.e., at the stationary points, both terms have the

same phase and opposite polarity". This is addressed in detail

in Appendix A. Therefore, using Eqs. !19" and !20" in Eq.

!18", we have

1

!

G !r ,r " = #G !r ,r"" G !r ,r"

(

S A B S A 0 B

i$#

r!!V

t

!

+ G !r ,r"" G !r ,r"$ · dS. !21"

0 B S A

FIG. 3. Cartoon illustrating case II. The medium conﬁguration

in this case is the same as for case I !Fig. 2", but now one of the

Since this equation is not affected by any changes to !V ,

b receivers at r is placed inside the perturbation volume P. Solid

B

this result is equally valid for P!V as in Fig. 2!b". This is

arrows illustrate stationary paths of reference waves and the dashed

one of the key results in this paper. For P!V, the results in

arrow illustrates the path of a scattered wave. Here r illustrates a

1

Eqs. !19" and !20" do not hold; by inserting Eq. !21" in Eq.

source position that yields a stationary contribution to the integrand

!18" we obtain the identity in Eq. !21".

036605-5VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

2

1 k

" = = . !24"

2 2

#c #$

For a homogeneous 1D medium with wave number k the

0

Green’s function solution of expression !23" is given by

#c

0

ik *z−z *

0 0

G !z,z " = e . !25"

0 0

2

For the particular case of a 1D medium, the surface integral

in Eq. !18" reduces to two end point contributions and the

volume integral becomes a line integral. With Eq. !24", Eq.

!18" under conditions set by Eq. !17" is given in one dimen-

sion by

FIG. 4. Schematic representation of case III, where P%V, i.e.,

G !z ,z " = S !z ,z " + S !z ,z " + V!z ,z ", !26"

the medium perturbation occupies all of the volume V. As in Fig. 2, S A B − A B + A B A B

solid and dashed arrows denote unperturbed waves and ﬁeld pertur-

with S !r ,r " as the contribution of a source above the re-

− B A

bations, respectively.

ceivers,

paths, prescribed by reference waves, are not affected by the

2

!

medium perturbation. Therefore, all integrals in Eq. !14"

S !z ,z " = G !z ,z "G !z ,z ", !27"

− A B S A − B −

0

#c

must always be evaluated. Another such example is case III 0

in Fig. 4, where the perturbations occur over the entire vol-

S !z ,z " as the contribution of a source below the receivers,

+ A B

ume, i.e., P%V #see condition !i"$ in Eq. !17".

2

!

IV. ANALYTIC EXAMPLE: 1D LAYERED MEDIA

S !z ,z " = G !z ,z "G !z ,z ", !28"

+ A B S A + B +

0

#c

0

As an example of case I #Fig. 2; deﬁned via Eq. !17"$ we

and V!z ,z " as the 1D volume integral,

present an acoustic one-dimensional model !Fig. 5" with a

A B

constant wave speed c and wave number k , except in a

0 0 H

i

2 2 !

layer of thickness H where the wave number is given by k .

1 V!z ,z " = !k − k "G!z ,z"G !z ,z"dz. !29"

A B ( A B

0 1 0

1 1

#$

This deﬁnes V=)z!R *z!#z ,z $+, P=)z!R *z!#0,H$+,

0

− +

2

2

and V!z"=k −k ∀z!P. It follows from the ﬁeld equations

0

The contributions of these different terms are sketched in

#e.g., Eq. !1"$ that for a 1D model with constant mass density

Fig. 5.

# the pressure satisﬁes

We ﬁrst consider the case in which the two receivers are

2

d p

located above the layer !z (0, z (0". The three contribu-

2 1 1 A B

+"#$ p = i$#q, z!R , p!C . !23"

2

tions to the perturbed Green’s function are sketched in panels

dz

!a"–!c" in Fig. 5. As shown in Appendix B the contribution

In this wave equation" is given by

from the source above the layer #Fig. 5, panel !a"$ gives the

perturbed Green’s function,

S !z ,z " = G !z ,z ". !30"

− A B S A B

This means that the contribution of this boundary point suf-

ﬁces to give the perturbed Green’s function. Note that the

perturbed Green’s function accounts for all reverberations

within the layer, as well as for the velocity change in the

layer. This demonstrates, in one dimension, the result in Eq.

!21". As with Eq. !21", the result in Eq. !30" holds regardless

of where the bottom source z is positioned, i.e., whether

+

P!V or P$V. It follows from a comparison of expressions

!26" and !30" that the contributions of S and V cancel,

+

FIG. 5. Location of the receiver coordinates z and z and the

S !z ,z " + V!z ,z " = 0. !31"

A B

+ A B A B

source coordinates for the example of the one-layer model. This

We show in Appendix B that this equality is indeed satisﬁed

being a one-dimensional example of case I !Fig. 2", the medium

for the one-layer system considered here. This, in turn, dem-

perturbation !in gray shading" is compactly supported in the interval

onstrates the result in Eq. !22".

#0,H$ where the jump in wave number is given by k −k . S , S ,

1 0 − +

Next consider sources on opposite sides of the layer

and V denote the 1D contributions of the top source, bottom source,

!z (0, z ’H" as sketched in panels !d"–!f" of Fig. 5. We

and line integral, respectively. The three leftmost vertical lines rep-

A B

show in Appendix B that now the source under the layer

resent the case where both receivers lie above the perturbations,

while panels !d"–!f" denote the case where there is a receiver on #panel !e" of Fig. 5$ sufﬁces to give the perturbed Green’s

either side of the perturbation. function,

036605-6REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"

S !z ,z " = G !z ,z ". !32"

+ A B S A B

We show in Appendix B that now the contributions S and V

−

cancel,

S !z ,z " + V!z ,z " = 0, !33"

− A B A B

which is, of course, required by Eq. !26". This result is in fact

the same as in Eq. !21" if only r were beneathP in Fig. 2!b"

B

and then the contributing surface would be !V instead of

b

!V . Since z and z are now in opposite sides ofP, Eqs. !32"

t B A

FIG. 6. Illustrations of energy considerations for extracting scat-

and !33" also demonstrate that the general results in Eqs. !21"

tered waves from random volume sources in V. To particularly

and !22" hold regardless of the position of the observation

highlight that equipartitioning is not a requirement for the retrieval

points r . It is interesting to note that the end point contri-

A of scattered waves; we use the medium conﬁguration of case I !Fig.

bution S satisﬁes

− 2". Panel !a" represents the case where energy is purely out going

!indicated by solid arrows" from P; this is the case for scattering in

−ik !z +z "

0 A B

S !z ,z " , e . !34"

− A B lossless media or when Im)V!r"+ and Im)" !r"+ are nonzero only

0

for r!P. In the case of general attenuative materials, depicted in

Note that a change in the choice of the coordinate system

!b", where Im)" !r"+"0 ∀r!V, there is an exchange of in- and

0

alters the phase of this term. This contribution therefore cor-

out-going energies on the boundary !P.

responds to an unphysical arrival with an arrival time that is

determined by the average position of the receivers. In

2

higher dimensions, this can also be observed by inspecting

G !r ,r "*R!$"* =− i$ *"!r ,$"%!r − r "

((

S A B 1 1 2

the volume terms in Eqs. !14" and !16". An improper cancel-

lation of this contribution with the volume term V would lead

2 !

)*R!$"* G!r ,r "G !r ,r "dV dV =

1 A 2 B 1 2

0

to unphysical arrivals in the extracted perturbed Green’s

function. It has been noted earlier that an inadequate source

− i$ G!r ,r "q!r "dV

/( 1 A 1 1

distribution may lead to unphysical arrivals in the extracted

Green’s function #26,28,29$.

!

) G !r ,r "q!r "dV . !38"

0( 12

0 2 B 2 2

V. RETRIEVING G FROM RANDOM SOURCES IN V:

S

ENERGY CONSIDERATIONS

Using the deﬁnitions in Eq. !37", Eq. !38" yields

Consider Eq. !15", i.e.,

− i$

!

G !r ,r " = -p!r "p !r ".. !39"

S B A A B

2 0

*R!$"*

!

G !r ,r " =− i$!" −""G!r ,r"G !r ,r"dV.

(

S A B 0 A 0 B

r!V

This equation shows that the perturbation response between

!35" r and r can be extracted simply by cross correlating the

B A

perturbed pressure ﬁeld observed at r with the unperturbed

A

When Dirichlet and/or Neumann boundary conditions apply

pressure measured at r . This cross correlation must be com-

B

2

#see derivation of Eq. !15"$, the pressure observed at any

pensated for the spectrum *R!$"* and multiplied by i$ !i.e.,

given observation point r is given by

o differentiated with respect to time".

Equation !39" is useful in understanding the energy parti-

tioning requirements for the reconstruction of the desired

p!r " = G!r ,r"q!r"dV !36"

(

o o

scattered-wave response. Let us consider, for example, Eq.

!39" for the conﬁguration of case I #Fig. 2, Eq. !17"$. In that

and likewise for unperturbed waves. q is the source term in

case, according to Eq. !37", the volume sources that are lo-

Eq. !1". Next we consider random uncorrelated sources dis-

cally proportional to the medium perturbation are restricted

tributed through space, such that

to P. This results in a nonzero net ﬂux that is outgoing en-

ergy at the boundary of P #we illustrate this in Fig. 6!a"$. As

! 2 d

-q!r ,$"q !r ,$". =*"!r ,$"%!r − r "*R!$"* , r !R ,

1 2 1 1 2 1,2

a consequence, there are also preferred directions of energy

ﬂux at the observation points r . This situation is com-

!37" A,B

pletely different than the condition of equipartitioning re-

2

where *"=" −" and *R!$"* is the power spectrum of a quired for the reconstruction of either G or G #7,25$, which

0 0

random excitation function and -·. denotes an ensemble av- requires that the total energy ﬂux within any direction at the

erage. Note from Eq. !37" that the source intensity is propor- receivers be equal to zero. To describe scattering, the ﬂux at

the sensor acts as a source must be so that it radiates energy

tional to the local perturbation *" #i.e., ,V!r"$ at every

2

source position. We then multiply Eq. !35" by *R!$"* to only toward the position of the scatterers. If the scatterers are

spatially restricted and located away from a sensor, then

obtain

036605-7VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

when acting as pseudosource this sensor only needs a limited +i Im!"−" "$ !where Re and Im denote real and imaginary

0

radiation aperture to fully reconstruct the scattered ﬁeld. This components, respectively", we obtain

explains why in the examples of case I #e.g., Eqs. !21", !30",

1

!

and !32"$ the full scattered ﬁeld is retrieved with a ﬁnite

G !r ,r " = #G !r ,r"" G !r ,r"

S A B ’ B S A

0

source aperture as long as the sensor acting as a source lies i$#

r!!V

between the physical sources and the scatterers.

!

− G !r ,r"" G !r ,r"$ · dS

S A B

0

1

VI. SCATTERING IN ATTENUATIVE MEDIA !

+ G!r ,r"Re)V!r"+G !r ,r"dV

( A B

0

i$#

r!V

To incorporate energy losses in wave propagation and

scattering, we take" !r","!r"!C #e.g., in Eq. !1"$ #30$. By

0 1

!

+ G!r ,r"Im)V!r"+G !r ,r"dV

(

A 0 B

using this in Eq. !8", Eq. !9" becomes

$#

r!V

A B! B! A A B!

#p v + p v $ · dS = p q dV

!

’ (

S 0 0 S S 0

− 2$ Im)" +G!r ,r"G !r ,r"dV. !41"

( 0 A B

0

r!!V r!V

r!V

! B!

A

The ﬁrst volume integral in Eq. !41" yields the volume inte-

+ i$!" −""p p dV,

(

0 0

r!V gral in Eq. !14", while the other volume integral accounts for

scattering attenuation. Note that in attenuative media, even if

!40"

there is no perturbation !i.e., V=0", the last volume integral

!

where now we have" instead of simply" #Eq. !9"$. Then,

in Eq. !41" is nonzero. This case is analyzed by Snieder #30$.

0 0

using Green’s functions #Eqs. !10" and !11"$ and deﬁning the

Let us revisit case I #Fig. 2 and Eq. !17"$, but now con-

2

complex scattering potential as V!r"=$ ##Re!"−" " sider it in attenuative media, i.e.,

0

d

!i" Re)V!r"+" 0 Im)V!r"+" 0; only for r!P; P!R

!ii" sing supp#Re)V!r"+$" 0; !i.e., V generates backscattering"

!iii" r "P; !i.e., perturbations away from receiver acting as source"

B

& d

!iv" " !r", # !r"! C !R "; !i.e.,smooth background"

0 0

!42"

−1

!v" #iG !r,r "$ " G !r,r " · n!r "’ 0; for !r,r "!!V or !V !i.e., outgoing reference waves"

0 s 0 s s s b t

−1

for !r,r "!!V !i.e., ingoing scattered waves"

#iG !r,r "$ " G !r,r " · n!r "( 0;

s b

S s S s s

d

!vi" Im)" !r"+ = 0, ∀ r!R ; or, !i.e., background is lossless"

0

!vi!" Im)" !r"+" 0; only for r!P; !i.e., background attenuation is restricted to P"

0

Next, under the same arguments as those used to derive Eqs.

1

! !

#G !r ,r"" G !r ,r"− G !r ,r"" G !r ,r"$ · dS

!19"–!21", it immediately follows that, for P$V #Fig. 2!c"$, ( B S A S A B

0 0

i$#

r!!V

b

1

!

= G!r ,r"Re)V!r"+G !r ,r"dV

(

A 0 B

1

! i$#

r!V

0 = G!r ,r"Re)V!r"+G !r ,r"dV

(

A B

0

i$#

r!V

1

!

+ G!r ,r"Im)V!r"+G !r ,r"dV

( A B

0

1

!

$#

r!V

+ G!r ,r"Im)V!r"+G !r ,r"dV

(

A 0 B

$#

r!V

!

− 2$ Im)" +G!r ,r"G !r ,r"dV. !44"

(

0 A 0 B

!

r!V

− 2$ Im)" +G!r ,r"G !r ,r"dV !43"

(

0 A B

0

r!V

Thus the general result of Eq. !21", discussed in Sec. V, is

also valid for attenuative scattered waves regardless of the

choice of conﬁgurations for !V or r !Fig. 2". So just as in

and that therefore Eqs. !20" and !21" are also valid for scat-

b A

tered waves in attenuative media. By extension to when lossless media, it is possible to retrieve the full scattered

response generated by soft and/or attenuative targets by cross

P!V in case I #Fig. 2!b"$, it is also true that

036605-8REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"

correlation of scattered and reference waves over a limited

source aperture.

To understand why the result above holds for attenuative

media, consider applying homogeneous Dirichlet or Neu-

mann conditions on !V in Eq. !41", which yields

1

!

G !r ,r " = G!r ,r"Re)V!r"+G !r ,r"dV

(

S A B A B

0

i$#

r!V

1

!

+ G!r ,r"Im)V!r"+G !r ,r"dV

(

A 0 B

$#

r!V

FIG. 7. !Color" Application of scattering reciprocity to acoustic

!

waves recorded on the ocean bottom. The cartoon in !a" relates the

− 2$ Im)" +G!r ,r"G !r ,r"dV. !45"

(

0 A B

0

speciﬁc case of ocean-ﬂoor seismology with the conﬁguration of

r!V

case I !Fig. 2". Panel !b" shows the perturbed acoustic wave speed

We now consider random volume sources similar to Eq. !37"

model used in the numerical experiment. In the model, the solid

but now described by

dotted line at 0.2 km depth represents the instrumented ocean bot-

tom, the open dotted line depicts the positions of physical sources,

! 2

-q!r ,$"q !r ,$". = Q!r "%!r − r "*R!$"* . !46"

1 2 1 1 2

and the triangle represents the location of the pseudosource in the

numerical examples. Note that, in the model in !b", the perturba-

Note that at every point in the volume, the quantity Q!r"

!

tions in P consist of the scatterers and interfaces located below the

=*" !r"=" !r"−"!r" in Eq. !46" describes sources which

!

0

depth of 0.3 km. The color bar portrays model wave speeds in km/s.

are locally proportional to !i" Re)V+, !ii" Im)V+, and !iii"

Im)" +, respectively. Through a derivation analogous to Eq.

0

!38", Eq. !45" gives

propagate in the experiment in Fig. 7, the recorded data con-

− i$ tain not only the desired subsurface scattered waves but also

!

G !r ,r " = -p!r "p !r "., !47"

S B A A 0 B

2 the reverberations that occur between the ocean surface, the

*R!$"*

sea bottom, and subsurface scatterers. These reverberations

same as in Eq. !39". For the conditions deﬁning case I in

become a strong source of coherent noise in extracting infor-

attenuative media #Eq. !42"$, the result in Eq. !47" implies a

mation about the Earth’s interior. Here we show that the

ﬂux of outgoing energy at the boundary of P, same as in the

scattering-based reciprocity relations developed in this paper

lossless case #Fig. 6!a"$. As with lossless scattering, the re-

can be used to remove the effect of surface-related reverbera-

ceiver that acts as a pseudosource needs only a limited ra-

tions from ocean-bottom seismic data, thus facilitating the

diation aperture to retrieve the full attenuative scattered-

retrieval of information associated only with subsurface scat-

wave response for case I; this is why the limited source

tered waves.

aperture used in Eq. !21" also accounts for attenuative scat-

Scattered waves described by reciprocity relations such as

tering. This scenario is no longer true, however, if the back-

in Eq. !18" satisfy boundary and initial conditions imposed

ground is attenuative, i.e., if conditions !vi" or !vi " in Eq.

! on!V !or in case I in!V #!V ", but it can be used to relate

b t

!42" do not hold. In that case, the result in Eq. !47" requires

different wave states that have varying material properties

the ignition of volume sources everywhere that are locally

and/or boundary conditions outside ofV #2,3$. In the particu-

proportional to the background loss parameters. This leads to

lar case of ocean-bottom seismics, the reciprocity relation in

an interchange of energy through the boundary of P, as de-

Eq. !18" can relate scattered waves in the presence of the

2

picted in Fig. 6!b". This in turn implies that, although energy

ocean’s free surface with waves without the free surface.

equipartitioning is still not a necessary requirement, the cor-

Note that the result in Eq. !21" is approximate for the case in

rect pseudoresponse between receivers cannot be retrieved

Fig. 7 because it violates condition !v": this leads to the

with a limited radiation aperture; consequently, Eq. !21"

incomplete cancellation of terms necessary for Eq. !20" to

would no longer hold.

hold. Furthermore, dipole acoustic sources are typically not

available in ocean-bottom seismic experiments. However,

VII. APPLICATION EXAMPLE: OCEAN-BOTTOM many such experiments do measure dual ﬁelds, i.e., pressure

SEISMICS and particle velocity, at the sea bottom. Since in the given

experiment the source surface is ﬂat and horizontal, i.e., n

Here we discuss the application of scattering reciprocity

=)0,0,n + ∀r!!V , then v!r,r "·dS=v !r,r "dS. In

3 t B,A i=3 B,A

to seismic data acquired on the ocean bottom. A general con-

the absence of vertically oriented dipole sources on !V , we

t

cept of ocean-bottom seismic data acquisition is shown in obs

replace them by v which is the response of monopole

i=3

Fig. 7. There, active physical sources are placed on !V and

t

sources observed in the vertical component of the particle

sensors are positioned on the seaﬂoor. The objective of

velocity ﬁeld at the ocean bottom. This gives, after Eq. !21",

ocean-bottom seismic experiments is to characterize the scat-

tering potential in the subsurface !i.e., in P; Fig. 7" from the

2

recorded scattered waves. Since the surface of the ocean acts

The term “free-surface” indicates that homogeneous Dirichlet

as a perfectly reﬂecting boundary for acoustic waves that conditions apply on the ocean surface.

036605-9VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

Offset (km) Offset (km) Offset (km) Offset (km)

0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0

0.2 0.2 0.2 0.2

0.4 0.4 0.4 0.4

0.6 0.6 0.6 0.6

0.8 0.8 0.8 0.8

(a) (b) (c) (d)

FIG. 8. Comparisons of true scattered-wave responses with pseudosource responses obtained by cross-correlating reference and scattered

waves. The true scattered-wave responses for a physical source at !0.3 km, 0.2 km" #see Fig. 7!b"$ are displayed in !a" modeled with a free

surface !at z=0 km" and in !b" where it is modeled without a free surface. The responses in !c" and in !d" correspond to pseudosources

retrieved via cross correlations. The result in !c" is obtained with Eq. !48", while !d" results from applying Eq. !49". It is important to note

that the input data to both !c" and !d" were modeled with a free surface.

are measured by a line of sensors on the water bottom #i.e., at

obs!

G !r ,r " 3 F!$"#p !r ,r"v !r ,r"

(

S A B S A B z=0.2 km; Fig. 7!b"$ positioned at every 2 m. With this ex-

3,0

r!!V

t

periment conﬁguration, we model the acoustic responses in

! obs

both the reference and perturbed models. All of the data used

+ p !r ,r"v !r ,r"$dS, !48"

B A

0 3,S

in for retrieving the scattered-wave Green’s functions be-

tween ocean-bottom sensors are modeled with free-surface

where F!$" is a signal-shaping ﬁlter that accounts for the

!i.e., Dirichlet" boundary conditions on the top of the model.

imprint of the source-time excitation function. Dipole

In Figs. 8!c" and 8!d" we show the result of extracting the

sources on !V can only be exactly replaced by observed

t

scattered-wave response between receivers using Eqs. !48"

particle velocities on the seaﬂoor if the surrounding medium

and !49", respectively. In both ﬁgures, the panels represent

was homogeneous. Therefore, using the observed quantities

obs

the responses recorded at all receivers !i.e., for varying r ",

A

v in Eq. !48" introduces errors in retrieving G !r ,r ".

3 S A B

excited by a pseudosource synthesized in ﬁxed receiver at

Because the material heterogeneity in our experiment is as-

r =!0.3 km,0.2 km" #triangle in Fig. 7!b"$. While Figs.

B

sociated with the scattering potential in P !Fig. 7", the errors

8!c" and 8!d" clearly show that the responses obtained via

obs

introduced by replacing dipole sources with v are of

3

Eqs. !48" and !49" are different, it is important to note that

higher order in the scattered waves !i.e., they will be rela-

the input ﬁeld quantities used for evaluating the integrands

tively weak in amplitudes". Most previous applications of

satisfy the same boundary conditions. On the other hand, the

retrieving inter-receiver Green’s functions from seismic data

responses of actual sources placed at !0.3 km, 0.2 km" de-

rely on the cross correlations of pressure ﬁelds only, i.e.,

picted in Figs. 8!a" and 8!b" satisfy different boundary con-

ditions: the pressure ﬁeld in Fig. 8!a" satisﬁes G !r"=0 at

0,S

2

! the sea surface #i.e., free-surface conditions; same as the in-

G !r ,r " 3 F!$"p !r ,r"p !r ,r"dS, !49"

S A B ( S A B

0

put ﬁelds for Figs. 8!c" and 8!d"$, while for the response in

#c

r!!V

t

Fig. 8!b" G !r""0 on the ocean surface. The response ob-

0,S

which assumes a far-ﬁeld or Sommerfeld radiation boundary tained by Eq. !48" #Fig. 8!c"$ approximates that of Fig. 8!b",

condition, e.g., #6$. In the example we present here we show whereas the response generated with Eq. !49" #Fig. 8!d"$ is

and discuss the differences of using Eqs. !48" and !49" for close to that of Fig. 8!a". In replacing the dipole sources

the extraction of the multiple-free scattered-wave response

required by Eq. !21" by the vertical component of particle

between ocean-bottom seismometers. velocity in Eq. !48", we achieve an effective cancellation

The two-dimensional numerical simulation is done on the

between in- and out-going waves at !V that results in ap-

t

model shown in Fig. 7!b". This model represents the per- proximating the scattered-wave response without the free-

turbed medium; the unperturbed medium consists only of the

surface condition present in the original experiment. On the

0.2-km-deep water layer and a homogeneous half-space with other hand, by cross correlating only pressure scattered and

a constant wave speed of 1800 m/s. The medium perturba-

reference waves #Eq. !49"$, we assume that there are only

tion thus consists of all scatterers and interfaces lying deeper out-going waves at !V and thus in- and out-going terms do

t

than 0.3 km #Fig. 7!b"$. Density is kept constant at not cancel. Consequently, when using Eq. !49" we retrieve

3 3

10 kg/m . We model the data using a ﬁnite-difference so- scattered waves that approximate the true perturbations in

lution to the acoustic wave equation #e.g., Eq. !1"$. In the the presence of free-surface boundary conditions #compare

numerical experiment, pressure !i.e., monopole" sources are Figs. 8!a" and 8!d"$.

placed on a 0.01-km-deep horizontal line with a constant With this numerical example we demonstrate that our for-

lateral spacing of 4 m; pressure and particle velocity ﬁelds mulations of scattering-based reciprocity can be used to ex-

036605-10

Time (s)

Time (s)

Time (s)

Time (s)REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"

tract scattered waves between receivers in ocean-bottom note that these results hold both for lossless and attenuative

seismic experiments. Moreover, we show that by using dif- scattering.

ferent combinations of single- or dual-ﬁeld measurements we In this paper we present a direct application of scattering

extract scattered ﬁelds that satisfy different boundary condi- reciprocity to ocean-bottom seismic data, where we retrieve

tions. This is a particularly important step in isolating or subsurface scattered waves from ocean-bottom receivers

separating the reverberations caused by the water surface without the interference of reverberations generated by the

from ocean-bottom seismic data. water surface. Other applications of the scattering reciprocity

relations to retrieving scattered signals have been proposed

in #29,32,33$. In the context of retrieving scattered waves by

VIII. DISCUSSION AND CONCLUSION

cross correlation, the theory we discuss also draws experi-

mental validation from the work of other authors. In particu-

In this paper, we present a suite of integral reciprocity

lar, we point out the studies performed by Bakulin and Cal-

equations for acoustic scattering that can be useful both for

vert #16$ and by Mehta et al. #34$, with their so-called virtual

theoretical considerations and for applications in retrieving

source method. Their methods explicitly correlate transmis-

scattered waves via correlations and possibly in imaging or

sion and reﬂection responses to extract desired scattered

inversion of scattered ﬁelds.

waves and directly verify our results. Note that although

A fundamental result in this paper is that the retrieval of

most of the examples cited here come from the ﬁeld of geo-

scattered waves by cross correlations or cross convolutions

physics, our results are immediately applicable to other ﬁelds

does not necessarily rely on a closed surface integral or on

in acoustics such as physical oceanography, laboratory, and

invoking energy equipartitioning. This is an important differ-

medical ultrasonics, and nondestructive testing.

ence between the work we present here and previous work in

While the derivations and examples presented here

the ﬁeld of Green’s function retrieval from diffuse-wave cor-

heavily focus on the application of scattering-based reciproc-

relation #9,17,31$ or from correlation of deterministic wave

ity to retrieving scattered responses by cross correlations, we

ﬁelds #6,25$, which do require energy equipartitioning. Most

point out some possible applications to inverse problems.

previous studies show that equipartitioning of energy is nec-

One such application is the use for the exact form of the

essary to recover the superposition of the causal and anti-

correlation-type representation theorems for the calculation

causal wave ﬁelds G or G !i.e., unperturbed or perturbed".

0

of Fréchet derivatives #35$, which consist of the partial de-

Since for scattered ﬁelds equipartitioning is not a necessary

rivatives of the wave ﬁeld perturbations with respect to the

requirement, our expressions isolate the wave ﬁeld perturba-

!

medium perturbations. These derivatives can be directly de-

tions G separately from its anticausal counterpart G . More-

S

S

rived from the theorems we provide here. These derivatives

over, for systems that are invariant under time reversal,

are important for the computation of sensitivity kernels used

Green’s function retrieval by wave ﬁeld cross correlations

in wave form inversion #35,36$, in imaging #37$, or in for-

requires only a surface integration #6,25,31$, whereas the re-

mulations of wave-equation based tomography #36,38$. Still

trieval of the perturbations G from correlations of wave

S

in the context of inverse scattering #5,37$, the theory we

ﬁeld perturbations with unperturbed wave ﬁelds requires ad-

present here is used in #39$ for establishing formal connec-

ditional volume integrals. Our analysis shows that, in fact,

tions between different approaches in imaging such as seis-

these volume terms counteract the contributions of closed

mic migration #40,41$, time-reversal methods #42,43$, and

surface terms, which reafﬁrms that, for arbitrarily spatially

image-domain inverse scattering #41,44$.

varying scattering potentials, the retrieval of scattered ﬁelds

Apart from imaging applications, our results !both in

relies on uneven energy partitioning.

terms of retrieving wave ﬁeld perturbations and for estimat-

This requirement of uneven radiation for the retrieval of

ing medium perturbations" can be used for monitoring tem-

scattered waves can be advantageous for certain experiment

poral changes in the medium. In geoscience, this could be

conﬁgurations. In the case of scattered waves generated by

applied to remotely monitoring the depletion of aquifers or

remote perturbations, we demonstrate that the scattered ﬁeld

hydrocarbon reservoirs or monitoring the injection of CO

propagating between receivers is fully retrieved by correlat- 2

for carbon sequestration. In materials science, our results can

ing scattered and reference waves generated by sources in an

be used to monitor material integrity with respect, for ex-

open surface. Again, previous general formulations of

ample, to temporal changes in temperature or changes due to

Green’s function retrieval #7,24,25$ state that sources must

crack formation. The detection of earthquake damage is a

surround the receivers to correctly retrieve, via cross corre-

potential application in the ﬁeld of structural engineering.

lations, the waves that propagate between receivers. In the

Within medical imaging applications, our expressions can be

absence of a closed source aperture, the retrieved responses

tailored, for instance, to observe the evolution of living tissue

are prone to dynamic distortions and artifacts #26,29$. This

!e.g., transplants and tumors" from a series of time-lapse ul-

becomes a limitation for the retrieval of receiver responses

trasonic measurements.

by correlations in experiments where surrounding the me-

dium with sources is not practical. If, however, the retrieval

of scattered waves is the objective, then our results show that

ACKNOWLEDGMENTS

the scattered ﬁeld can be accurately retrieved with a limited

source array !for the conﬁguration in Fig. 2". This is an im- This research was ﬁnanced by the NSF !Grant No. EAS-

portant experimental advantage brought by the analysis of 0609595" by the sponsors of the Consortium for Seismic

scattering-based reciprocity. Furthermore, it is important to

Inverse Methods for Complex Structures at the Center for

036605-11VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

FIG. 10. Cartoons representing the conditions required for the

bottom-surface integral to vanish in the case of Eq. !20". Panel !a"

FIG. 9. Illustration of stationary points on the bottom surface states that ingoing reference waves due to sources on !V must be

b

!V that yield physical contributions to scattered waves that propa- absent, whereas !b" indicates that there should be no outgoing scat-

b

gate between the observation points. tered waves excited by sources on !V .

b

Wave Phenomena and by ION Geophysical, GXT Imaging +i*cos,!x"*$/c!x", where c!x" is the local velocity at !V

b

Solutions. We thank Evert Slob, Kees Wapenaar, and Deyan and ,!x" is the local angle between the ray and the normal

Draganov !all TU Delft" for discussions that greatly contrib- on !V . The minus sign relates to waves traveling into V,

b

uted to this paper. while the plus sign relates to waves traveling out of V. By

the exact same reasoning as #6$ it follows that at the station-

ary source locations on !V the absolute values of the co-

b

APPENDIX A: CONDITIONS FOR A VANISHING

sines of the ray angles are the same for G and G . That

S,A 0,B

INTEGRAL OVER&V

b

means that contributions of the terms in Eq. !A3" with prod-

To determine the situation when the surface integral for

ucts of in- and out-going Green’s functions give exactly op-

the bottom surface in Eq. !20" vanishes, we ﬁrst observe that posite contributions. Therefore these “cross” terms do not

in general this surface integral cannot vanish. For example,

contribute to the surface integral, leaving the surface integral

consider the case when there is a free surface present above as

the bottom surface!V !Fig. 9". Then there will be stationary

b

1

! !

sources on the bottom surface that contribute to the construc-

)G " G − G " G + · dS

(

S,A S,A

0,B 0,B

tion of the scattered ﬁeld with a source at location r . The i$#

B

!V

b

drawn propagation paths in Fig. 9 are the outermost paths

2 ! !

that are still needed to illuminate the scattering region with in in out out

= )G " G + G " G + · dS. !A4"

(

S,A 0,B S,A 0,B

sources on the surface !V , and indeed all the sources in i$#

b !V

b

between s and s on the integration surface give station-

left right

From Eq. !A4" it is easy to see when the surface integral

ary contributions to the surface integral.

vanishes. The only meaningful situations to consider are the

To see in which special cases the surface integral does

cases

vanish, we follow #6$ and decompose the wave ﬁeld into

in out

!i" G =0 and n ˆ ·"G =0 and

in-going and out-going waves of the volume V. That is, we S,A 0,B

out in

ˆ

!ii" G =0 and n·"G =0,

assume

S,A 0,B

where n ˆ is the outward pointing normal on !V . We are

b

in out

G = G + G , !A1"

interested in analyzing when the surface integral vanishes if

0 0 0

the surface !V is above the perturbation volume P. In this

b

in out

situation case !i" is not really relevant, as then there would be

G = G + G . !A2"

S S S

no energy scattering into the volume V, which is not a com-

Using this in Eq. !20", it follows that

mon situation encountered. Therefore case !ii" provides the

relevant conditions when the surface integral vanishes. This

1

! !

)G " G − G " G + · dS means that there cannot be any scattered energy traveling

( S,A S,A

0,B 0,B

i$#

!V

outward of V through !V . That is, scattered energy is not

b

b

allowed to change propagation direction from into V to out

1 ! !

in out in out in out

of V above!V !or from up to down in case!V is horizon-

= )!G + G "!"G +"G "− !G + G "

( b b

0,B 0,B S,A S,A S,A S,A

i$#

!V

b tal". Moreover, the background wave ﬁeld cannot change

! ! propagation direction from out of V to into V below!V !or

in out b

)!"G +"G "+ · dS, !A3"

0,B 0,B

from down to up in case !V is horizontal". Both these con-

b

ditions are summarized in Fig. 10.

where we introduced the shorthand notation G

!S,0",!A,B"

=G !r ,r" with the subscripts !S,0" indicating either

!S,0" !A,B"

the scattered !S" or background !0" Green’s function, while

APPENDIX B: ANALYSIS OF THE SCATTERED-WAVE

the subscripts !A,B" denote the receiver location at either r

A

RESPONSES FOR THE ONE-LAYER MODEL

!A" or r !B". Following again #6$ and assuming that the

B

medium is locally smooth around !V , we can approximate In this appendix we derive Green’s function extraction for

b

the 1D model of Fig. 5. Within every layer, the solution

the gradients by a multiplication of the Green’s function with

036605-12REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"

H

consists of the superposition of waves exp!+ikz", with k as

i #c 1 k

0 1

2 2

ik !z−H"−ik z

d#!z" 1 0 A

V!z ,z " = !k − k " ) 01 + 1e

( 4

B A 0 1

the wave number in each layer. Since =0, G!z,z "

0

dz

#$ 2 2D k

0 0

2 1

!C !R ". For a source above the layer !z (0" this leads to

0

the following exact Green’s function for z(0: k #c

1 0

−ik !z−H"−ik z −ik !z−z "

1 0 A 0 B

+ 1− e e dz. !B8"

0 1

5

k 2

0

#c #c i k k

0 0 1 0

ik *z−z * −ik !z+z "

0 0 0 0

G!z,z " = e + − sin k He ,

0 1

0 1

2 2 2D k k

0 1

Carrying out the z integration and rearranging terms gives

!B1"

#c 1 1 k k

0 0 1

ik !z −z "

0 B A

while for 0(z(H V!z ,z " =− e 1 + +

64 0 15

B A

2 2D 2 k k

1 0

#c 1 k

0 0

ik !z−H"−ik z

1 k k

1 0 0

0 1

G!z,z " = 1 + e −ik H −ik H

0 1

0 1

0

)!e − e " + 1− +

0 1

4 5

2 2D k

1

2 k k

1 0

#c 1 k

0 0

−ik !z−H"−ik z

1 0 0

+ 1− e !B2" −ik H ik H

0 1

0 1

)!e − e " . !B9"

7

2 2D k

1

and for z’H

The term between curly brackets satisﬁes

#c 1

0

ik !z−H−z "

0 0

G!z,z " = e , !B3"

0

1 k k

0 1

−ik H ik H −ik H ik H −ik H

2 D

0 1 1 1 1

#¯ $ = 2e − !e + e " + + !e − e "

0 1

2 k k

1 0

with

−ik H

0

= 2!e − D", !B10"

i k k

1 0

D = cos k H− + sin k H. !B4"

0 1

1 1

where expression !B4" is used in the last identity. Using this

2 k k

0 1

result gives

For z(0 the perturbed ﬁeld is given by the last term of

expression !B1", while for z’H the perturbed ﬁeld G =G #c 1

S 0

−ik H ik !z −z "

0 0 B A

V!z ,z " =− !e − D"e . !B11"

B A

−G follows by subtracting expressions !B3" and !25",

0 2 D

#c 1

0

ik !z−z " −ik H

0 0 0 A comparison with Eq. !B7" proves expression !31".

G !z,z " = e !e − D". !B5"

S 0

2

D

We next consider the situation where the receivers are on

opposite sides of the layer #panels !d"–!f" in Fig. 5$. The term

We ﬁrst compute the contribution S when both receivers

−

S #panel !e"$ follows by combining expressions !25", !28",

+

are above the layer #panel !a" of Fig. 5$. Inserting the last

and !B5" to give

term of expression !B1" and Eq. !25" into expression !27"

gives

2 #c 1 #c

0 0

ik !z −z " −ik H −ik !z −z "

0 + A 0 0 + B

S !z ,z " = e !e − D" e

+ B A

2 #c i k k

#c 2 D 2

0 1 0

0

S !z ,z " = −

0 1

− B A

#c 2 2D k k

0 0 1

1

#c

0

ik !z −z " −ik H

0 B A 0

= e !e − D". !B12"

#c

2 D

0

−ik !z +z " −ik !z −z "

0 A − 0 B −

)sin k He e

1

2

A comparison with expression !B5" shows that this equals

#c i k k

0 1 0

−ik !z +z " the ﬁeld perturbation #expression !32"$. The contribution

0 A B

= − sin k He . !B6"

0 1

1

2 2D k k from the other end point #panel !d" in Fig, 5$ follows by

0 1

combining expressions !25", !27", and !B1",

A comparison with the last term of expression !B1" shows

that S gives the perturbed Green’s function #expression

−

2 #c i k k

0 1 0

!30"$. The contribution from a source below the layer #panel S !z ,z " = −

0 1

− B A

#c 2 2D k k

0 0 1

!b" of Fig. 5$ follows by inserting expressions !25" and !B5"

into Eq. !28",

#c

0

−ik !z +z " −ik !z −z "

0 − A 0 B −

)sin k He e

1

2

ik !z −z "

0 + A

2 #c e #c

0 0

−ik H −ik !z −z "

0 0 + B

S !z ,z " = !e − D" e

+ B A

#c i k k

#c 2 D 2

0 1 0

0 −ik !z +z "

0 A B

= − sin k He .

0 1

1

ik !z −z " 2 2D k k

0 B A 0 1

#c e

0

−ik H

0

= !e − D". !B7"

!B13"

2 D

To get the volume term #panel !c" of Fig. 5$ we insert ex- The volume term #panel !f" of Fig. 5$ follows from combin-

ing expressions !25", !29", and !B2",

pressions !25" and !B2" into Eq. !29" to give

036605-13VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"

H 2

i #c 1 k i #c 1 2

0 0 0

2 2 2 2 −ik !z +z "

ik !z−H"−ik z

0 A B

1 0 A

V!z ,z " = !k − k " sin k He

V!z ,z " = !k − k " ) 01 + 1e 0 1

( 4 B A 1

B A 0 1 0 1

#$ 2 2D k

#$ 2 2D k

1

0 1

#c i k k

k #c 0 1 0

−ik !z +z "

0 0

0 A B

−ik !z−H"−ik z −ik !z −z"

1 0 A 0 B =− − sin k He . !B15"

0 1

+ 1− e e dz. !B14" 1

0 1

5

2 2D k k

0 1

k 2

1

Together with Eq. !B13" this proves Eq. !33".

Carrying out the z integration and using that$/k =c give

0 0

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036605-14

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