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
inversescattering 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 convolutiontype theorems can be manipulated to obtain scattering integrals that are analogous to the
LippmannSchwinger equation, the correlationtype 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
scatteredwave 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 onedimensional 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 threedimensional media by Wapenaar et al. #13$,
portant properties of wave propagation phenomena. Rayleigh
who used oneway reciprocity theorems in their derivations.
#1$ used a local form of an acoustic reciprocity theorem to
Green’s function retrieval by cross correlations has found
demonstrate sourcereceiver reciprocity. Timedomain 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$", the derivations based on
ogy, an important application of convolutiontype reciprocity
representation theorems have provided for generalizations
theorems lies in removing multiple reﬂections, also called
beyond lossless acoustic wave propagation to elastic wave
multiples, caused by the Earth’s free surface #3,4$. These
propagation and diffusion. More general forms of reciprocity
approaches rely on the convolution of singlescattered 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 inversescattering
diffusion, advection, and Schrödinger equations, among oth
methods #5$. The inversescatteringbased methodologies are
ers.
1
typically used separately from the representation theorem
In this paper, we derive reciprocity theorems for acoustic
approaches #3,4$ 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$, showing that the
for monitoring. For imaging, the unperturbed medium is as
cross correlations of waves recorded by two receivers can be
sumed to be so smooth that it does not generate reﬂected
used to obtain the waves that propagate between these re
waves, while discontinuities in the perturbation account for
ceivers as if one of them behaves as a source. These results
scattering. In monitoring applications, the perturbation con
coincide with other studies based on cross correlations of
sists of the timelapse changes in the medium. Although pre
diffuse waves in a medium with an irregular boundary #8$,
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 onedimensional !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$.
15393755/2009/80!3"/036605!14" 0366051 ©2009 The American Physical SocietyVASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"
Here we focus on deriving and discussing some of these
differences.
One particularly important aspect of studying scattering
based reciprocity lies in retrieving wave ﬁeld perturbations
from cross correlations #7,25$. 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 fullﬁeld 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
correlationtype 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 convolutiontype 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 righthand 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 onedimensional 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 stressstrain 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 stressstrain 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 timeharmonic 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 frequencydomain 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,$" describes the source
A correlationtype reciprocity theorem #2,3$ 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
0366052REPRESENTATION 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 correlationtype 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 $dV,
’ (
0 0 0 0 0 0 0 0
r!!V r!V density perturbations. Such modiﬁcation involves adding, to
the RHS of the equations, an extra volume integral whose
!5"
A
integrand is proportional to !# −#" and the wave ﬁelds v
0
B B!
where complex conjugates translate into timedomain cross
and v !or v " #3$.
0 0
correlations of ﬁeld parameters. For this reason, Eq. !5" is a
reciprocity theorem of the correlation type #2,3$.
Convolution and correlationtype reciprocity theorems for
III. SCATTERINGBASED REPRESENTATIONS AND
the perturbed wave states #e.g., Eq. !1"$ 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,$".
A,B
r!!V r!V
Using these deﬁnitions, the convolutiontype theorem in
Eq. !7" becomes
A B
+ i$!" −""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 convolutiontype reciprocity theorem for per
turbed media.
1
= #G !r ,r"" G !r ,r"
’
S A 0 B
The correlationtype 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 correlationtype 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$#
r!V
!i.e., no attenuation".
0366053VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"
FIG. 2. Schematic illustrations of conﬁgurations for case I. Medium perturbations are restricted to the subdomainP, which is placed away
from the observation points. By inﬁnitely extending the sides of !V, the closed surface integral can be replaced by an integral over r
!!V #!V , as portrayed in panels !a" and !b". In our discussion, we ﬁx the sets !V and P and have two choices for !V such that in !b"
b t t b
P!V and in !c" P$V.
Equation !13" is the integral equation known as the taking either timeadvanced #i.e., G !r ,r"; Eq. !13"$ or
0 B
!
LippmannSchwinger equation #27$, commonly used for
timereversed #i.e., G !r ,r"; Eq. !15"$ ﬁelds.
B
0
modeling and inversion or imaging in scattering problems.
The lefthand 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
righthand side of Eq. !13".
changing subscripts A by B!, and taking the complex conju
Next, we turn our attention to the correlationtype 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"$, from cross
correlations of G with G . Second, the proper manipulation
S 0
of unperturbed waves G and perturbations G in the inte
0 S
The surface integral here does not vanish under a Sommer
grands of Eqs. !14" and !16" allows for the separate retrieval
feld radiation condition, but with Dirichlet and/or Neumann
of causal and anticausal wave ﬁeld perturbations G !r ,r "
S A B
boundary conditions we get
in the frequency domain rather than their superposition.
Since the correlationtype representation theorems for G or
G #7,25$ result in the superposition of causal and anticausal
0
1 responses in the frequency domain, their timedomain 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 twosided
theorems. The theorems in Eqs. !14" and !16" recover the
timedomain ﬁeld perturbation response for either positive
which is similar to the LippmannSchwinger 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" onesided 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
0366054REPRESENTATION 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 crosscorrelating 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
bottomsurface 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 sourcereceiver conﬁgurations
i$#
r!V
with onedimensional analytic examples !see below".
In general, the volume integrals in Eq. !14" cannot be
Furthermore, if P$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".
0366055VASCONCELOS, 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 onedimensional 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$V. It follows from a comparison of expressions
!26" and !30" that the contributions of S and V cancel,
+
FIG. 5. Location of the receiver coordinates z and z and the
S !z ,z " + V!z ,z " = 0. !31"
A B
+ A B A B
source coordinates for the example of the onelayer model. This
We show in Appendix B that this equality is indeed satisﬁed
being a onedimensional example of case I !Fig. 2", the medium
for the onelayer system considered here. This, in turn, dem
perturbation !in gray shading" is compactly supported in the interval
onstrates the result in Eq. !22".
#0,H$ 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,
0366056REPRESENTATION 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
outgoing 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
scatteredwave 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!$"* to only toward the position of the scatterers. If the scatterers are
spatially restricted and located away from a sensor, then
obtain
0366057VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"
when acting as pseudosource this sensor only needs a limited +i Im!"−" "$ !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"$, it is also true that
0366058REPRESENTATION THEOREMS AND GREEN’s FUNCTION… PHYSICAL REVIEW E 80, 036605 !2009"
correlation of scattered and reference waves over a limited
source aperture.
To understand why the result above holds for attenuative
media, consider applying homogeneous Dirichlet or Neu
mann conditions on !V in Eq. !41", which yields
1
!
G !r ,r " = G!r ,r"Re)V!r"+G !r ,r"dV
(
S A B A B
0
i$#
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 oceanﬂoor 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
scatteringbased reciprocity relations developed in this paper
lossless case #Fig. 6!a"$. As with lossless scattering, the re
can be used to remove the effect of surfacerelated reverbera
ceiver that acts as a pseudosource needs only a limited ra
tions from oceanbottom seismic data, thus facilitating the
diation aperture to retrieve the full attenuative scattered
retrieval of information associated only with subsurface scat
wave response for case I; this is why the limited source
tered waves.
aperture used in Eq. !21" also accounts for attenuative scat
Scattered waves described by reciprocity relations such as
tering. This scenario is no longer true, however, if the back
in Eq. !18" satisfy boundary and initial conditions imposed
ground is attenuative, i.e., if conditions !vi" or !vi " in Eq.
! on!V !or in case I in!V #!V ", but it can be used to relate
b t
!42" do not hold. In that case, the result in Eq. !47" requires
different wave states that have varying material properties
the ignition of volume sources everywhere that are locally
and/or boundary conditions outside ofV #2,3$. In the particu
proportional to the background loss parameters. This leads to
lar case of oceanbottom 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 oceanbottom seismic experiments. However,
VII. APPLICATION EXAMPLE: OCEANBOTTOM 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 oceanbottom 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",
oceanbottom 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 “freesurface” indicates that homogeneous Dirichlet
as a perfectly reﬂecting boundary for acoustic waves that conditions apply on the ocean surface.
0366059VASCONCELOS, 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 scatteredwave responses with pseudosource responses obtained by crosscorrelating reference and scattered
waves. The true scatteredwave 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"$dS, !48"
B A
0 3,S
in for retrieving the scatteredwave Green’s functions be
tween oceanbottom sensors are modeled with freesurface
where F!$" is a signalshaping ﬁlter that accounts for the
!i.e., Dirichlet" boundary conditions on the top of the model.
imprint of the sourcetime 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
scatteredwave 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"$. While Figs.
B
sociated with the scattering potential in P !Fig. 7", the errors
8!c" and 8!d" clearly show that the responses obtained via
obs
introduced by replacing dipole sources with v are of
3
Eqs. !48" and !49" are different, it is important to note that
higher order in the scattered waves !i.e., they will be rela
the input ﬁeld quantities used for evaluating the integrands
tively weak in amplitudes". Most previous applications of
satisfy the same boundary conditions. On the other hand, the
retrieving interreceiver Green’s functions from seismic data
responses of actual sources placed at !0.3 km, 0.2 km" de
rely on the cross correlations of pressure ﬁelds only, i.e.,
picted in Figs. 8!a" and 8!b" satisfy different boundary con
ditions: the pressure ﬁeld in Fig. 8!a" satisﬁes G !r"=0 at
0,S
2
! the sea surface #i.e., freesurface conditions; same as the in
G !r ,r " 3 F!$"p !r ,r"p !r ,r"dS, !49"
S A B ( S A B
0
put ﬁelds for Figs. 8!c" and 8!d"$, while for the response in
#c
r!!V
t
Fig. 8!b" G !r""0 on the ocean surface. The response ob
0,S
which assumes a farﬁeld or Sommerfeld radiation boundary tained by Eq. !48" #Fig. 8!c"$ 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 multiplefree scatteredwave response
required by Eq. !21" by the vertical component of particle
between oceanbottom seismometers. velocity in Eq. !48", we achieve an effective cancellation
The twodimensional numerical simulation is done on the
between in and outgoing waves at !V that results in ap
t
model shown in Fig. 7!b". This model represents the per proximating the scatteredwave response without the free
turbed medium; the unperturbed medium consists only of the
surface condition present in the original experiment. On the
0.2kmdeep water layer and a homogeneous halfspace 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"$, we assume that there are only
tion thus consists of all scatterers and interfaces lying deeper outgoing waves at !V and thus in and outgoing terms do
t
than 0.3 km #Fig. 7!b"$. 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 ﬁnitedifference so scattered waves that approximate the true perturbations in
lution to the acoustic wave equation #e.g., Eq. !1"$. In the the presence of freesurface boundary conditions #compare
numerical experiment, pressure !i.e., monopole" sources are Figs. 8!a" and 8!d"$.
placed on a 0.01kmdeep 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 scatteringbased reciprocity can be used to ex
03660510
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 oceanbottom 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 dualﬁeld measurements we In this paper we present a direct application of scattering
extract scattered ﬁelds that satisfy different boundary condi reciprocity to oceanbottom seismic data, where we retrieve
tions. This is a particularly important step in isolating or subsurface scattered waves from oceanbottom receivers
separating the reverberations caused by the water surface without the interference of reverberations generated by the
from oceanbottom 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$, with their socalled virtual
theoretical considerations and for applications in retrieving
source method. Their methods explicitly correlate transmis
scattered waves via correlations and possibly in imaging or
sion and reﬂection responses to extract desired scattered
inversion of scattered ﬁelds.
waves and directly verify our results. Note that although
A fundamental result in this paper is that the retrieval of
most of the examples cited here come from the ﬁeld of geo
scattered waves by cross correlations or cross convolutions
physics, our results are immediately applicable to other ﬁelds
does not necessarily rely on a closed surface integral or on
in acoustics such as physical oceanography, laboratory, and
invoking energy equipartitioning. This is an important differ
medical ultrasonics, and nondestructive testing.
ence between the work we present here and previous work in
While the derivations and examples presented here
the ﬁeld of Green’s function retrieval from diffusewave cor
heavily focus on the application of scatteringbased reciproc
relation #9,17,31$ or from correlation of deterministic wave
ity to retrieving scattered responses by cross correlations, we
ﬁelds #6,25$, which do require energy equipartitioning. Most
point out some possible applications to inverse problems.
previous studies show that equipartitioning of energy is nec
One such application is the use for the exact form of the
essary to recover the superposition of the causal and anti
correlationtype representation theorems for the calculation
causal wave ﬁelds G or G !i.e., unperturbed or perturbed".
0
of Fréchet derivatives #35$, 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$, whereas the re
mulations of waveequation based tomography #36,38$. 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$, timereversal methods #42,43$, and
surface terms, which reafﬁrms that, for arbitrarily spatially
imagedomain 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$. This
!e.g., transplants and tumors" from a series of timelapse ul
becomes a limitation for the retrieval of receiver responses
trasonic measurements.
by correlations in experiments where surrounding the me
dium with sources is not practical. If, however, the retrieval
of scattered waves is the objective, then our results show that
ACKNOWLEDGMENTS
the scattered ﬁeld can be accurately retrieved with a limited
source array !for the conﬁguration in Fig. 2". This is an im This research was ﬁnanced by the NSF !Grant No. EAS
portant experimental advantage brought by the analysis of 0609595" by the sponsors of the Consortium for Seismic
scatteringbased reciprocity. Furthermore, it is important to
Inverse Methods for Complex Structures at the Center for
03660511VASCONCELOS, SNIEDER, AND DOUMA PHYSICAL REVIEW E 80, 036605 !2009"
FIG. 10. Cartoons representing the conditions required for the
bottomsurface integral to vanish in the case of Eq. !20". Panel !a"
FIG. 9. Illustration of stationary points on the bottom surface states that ingoing reference waves due to sources on !V must be
b
!V that yield physical contributions to scattered waves that propa absent, whereas !b" indicates that there should be no outgoing scat
b
gate between the observation points. tered waves excited by sources on !V .
b
Wave Phenomena and by ION Geophysical, GXT Imaging +i*cos,!x"*$/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 outgoing 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
ingoing and outgoing 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 SCATTEREDWAVE
the subscripts !A,B" denote the receiver location at either r
A
RESPONSES FOR THE ONELAYER 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
03660512REPRESENTATION 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
03660513VASCONCELOS, 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$/k =c give
0 0
#1$ J. W. S. Rayleigh, The Theory of Sound !Dover, New York, Hodgkiss, IEEE J. Ocean. Eng. 30, 338 !2005".
1878" !reprint 1945". #24$ R. L. Weaver and O. I. Lobkis, J. Acoust. Soc. Am. 116, 2731
#2$ A. de Hoop, J. Acoust. Soc. Am. 84, 1877 !1988". !2004".
#3$ J. T. Fokkema and P. M. van den Berg, Seismic Applications of #25$ R. Snieder, K. Wapenaar, and U. Wegler, Phys. Rev. E 75,
Acoustic Reciprocity !Elsevier, New York, 1993". 036103 !2007".
#4$ A. J. Berkhout and D. J. Verschuur, Geophysics 62, 1586 #26$ R. L. Weaver, Wave Motion 45, 596 !2008".
!1997". #27$ L. S. Rodberg and R. M. Thaler, Introduction to the Quantum
#5$ A. B. Weglein, F. V. Araújo, P. M. Carvalho, R. H. Stolt, K. H. Theory of Scattering !Academic Press, New York, 1967".
Matson, R. T. Coates, D. Corrigan, D. J. Foster, S. A. Shaw, #28$ R. Snieder, K. van Wijk, M. Haney, and R. Calvert, Phys. Rev.
and H. Zhang, Inverse Probl. 19, R27 !2003". E 78, 036606 !2008".
#6$ K. Wapenaar and J. Fokkema, Geophysics 71, SI133 !2006". #29$ I. Vasconcelos and R. Snieder, Geophysics 73, S115 !2008".
#7$ K. Wapenaar, E. Slob, and R. Snieder, Phys. Rev. Lett. 97, #30$ R. Snieder, J. Acoust. Soc. Am. 121, 2637 !2007".
234301 !2006". #31$ E. Larose, L. Margerin, A. Derode, B. van Tiggelen, M. Camp
#8$ O. I. Lobkis and R. L. Weaver, J. Acoust. Soc. Am. 110, 3011 illo, N. Shapiro, A. Paul, L. Stehly, and M. Tanter, Geophysics
!2001". 71, SI11 !2006".
#9$ R. L. Weaver and O. I. Lobkis, Phys. Rev. Lett. 87, 134301 #32$ I. Vasconcelos, R. Snieder, and B. Hornby, Geophysics 73,
!2001". S157 !2008".
#10$ N. M. Shapiro, M. Campillo, L. Stehly, and M. H. Ritzwoller, #33$ I. Vasconcelos, R. Snieder, S. T. Taylor, P. Sava, J. A. Chavar
Science 307, 1615 !2005". ria, and P. Malin, EOS Trans. Am. Geophys. Union 89, 349
#11$ R. Snieder, Phys. Rev. E 69, 046610 !2004". !2008".
#12$ J. F. Claerbout, Geophysics 33, 264 !1968". #34$ K. Mehta, A. Bakulin, J. Sheiman, R. Calvert, and R. Snieder,
#13$ K. Wapenaar, J. Thorbecke, and D. Draganov, Geophys. J. Int. Geophysics 72, V79 !2007".
156, 179 !2004". #35$ A. Tarantola, Inverse Problem Theory !Elsevier, Amsterdam,
#14$ K. G. Sabra, P. Gerstoft, P. Roux, W. A. Kuperman, and M. 1987".
Fehler, Geophys. Res. Lett. 32, L14311 !2005". #36$ J. Tromp, C. Tape, and Q. Liu, Geophys. J. Int. 160, 195
#15$ G. T. Schuster, F. Followill, L. J. Katz, J. Yu, and Z. Liu, !2005".
Geophysics 68, 1685 !2003". #37$ D. Colton and R. Kress, Inverse Acoustic and Electromagnetic
#16$ A. Bakulin and R. Calvert, Geophysics 71, SI139 !2006". Scattering Theory !SpringerVerlag, Berlin, 1992".
#17$ A. E. Malcolm, J. A. Scales, and B. A. van Tiggelen, Phys. #38$ M. Woodward, Geophysics 57, 15 !1992".
Rev. E 70, 015601!R" !2004". #39$ I. Vasconcelos, SEG Exp. Abstr. 27, 2927 !2008".
#18$ K. van Wijk, Geophysics 71, SI79 !2006". #40$ J. F. Claerbout, Imaging the Earth’s Interior !Blackwell, Cam
#19$ J. E. Rickett and J. F. Claerbout, The Leading Edge 18, 957 bridge, MA, 1985".
!1999". #41$ C. Stolk and M. de Hoop, Wave Motion 43, 579 !2006".
#20$ R. Snieder and E. Şafak, Bull. Seismol. Soc. Am. 96, 586 #42$ M. Fink, W. A. Kuperman, J.P. Montagner, and A. Tourin,
!2006". Imaging of Complex Media with Acoustic and Seismic Waves
#21$ D. Thompson and R. Snieder, The Leading Edge 25, 1093 !SpringerVerlag, Berlin, 2002".
!2006". #43$ J.P. Fouque, J. Garnier, G. Papanicolaou, and K. Solna, Wave
#22$ P. Roux, W. A. Kuperman, and NPAL Group, J. Acoust. Soc. Propagation and Time Reversal in Randomly Layered Media
Am. 116, 1995 !2004". !Springer, New York, 2007".
#23$ K. G. Sabra, P. Roux, A. M. Thode, G. L. D’Spain, and W. S. #44$ G. Beylkin, J. Math. Phys. 26, 99 !1985".
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