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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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