connection of scattering principles: a visual and

CWP-690

Connection of scattering principles: a visual andmathematical tour

Filippo Broggini & Roel SniederCenter for Wave Phenomena, Colorado School of Mines, Golden, CO 80401, USA

ABSTRACTInverse scattering, Green’s function reconstruction, focusing, imaging, and theoptical theorem are subjects usually studied as separate problems in differentresearch areas. We speculate that a physical connection exists between them be-cause the equations that rule these scattering principles have a similar functionalform. We first lead the reader through a visual explanation of the relationshipbetween these principles and then present the mathematics that illustrates thelink between the governing equations of these principles. Throughout this work,we describe the importance of the interaction between the causal and anti-causalGreen’s functions.

Key words: focusing, inverse scattering, Green’s function reconstruction,optical theorem, imaging

1 INTRODUCTION

Inverse scattering, Green’s function reconstruction, fo-cusing, imaging, and the optical theorem are subjectsusually studied in different research areas such as seis-mology (Aki and Richards, 2002), quantum mechanics(Rodberg and Thaler, 1967), optics (Born and Wolf,1999), non-destructive evaluation of material (Shull,2002), and medical diagnostics (Epstein, 2003).

Inverse scattering (Chadan and Sabatier, 1989;Gladwell, 1993; Colton and Kress, 1998) is the prob-lem of determining the perturbation of a medium (e.g.,of a constant velocity medium) from the field scatteredby this perturbation. In other words, one aims to recon-struct the properties of the perturbation (represented bythe scatterer in Figure 1) from a set of measured data.Inverse scattering takes into account the nonlinearityof the inverse problem, but it also presents some draw-backs: it is improperly posed from the point of view ofnumerical computations (Dorren et al., 1994), and it re-quires data recorded at locations usually not accessibledue to practical limitations.

Green’s function reconstruction (Wapenaar et al.,2005) is a technique that allows one to reconstruct theresponse between two receivers (represented by the twotriangles at locations RA and RB in Figure 2) fromthe cross-correlation of the wavefield measured at thesetwo receivers which are excited by uncorrelated sources

Scatterer

RS

Scattered field → Scatterer properties

Figure 1. Inverse scattering is the problem of determiningthe perturbation of a medium from its scattered field.

surrounding the studied system. In the seismic commu-nity, this technique is also known as either the virtualsource method (Bakulin and Calvert, 2006) or seismicinterferometry (Curtis et al., 2006; Schuster, 2009). Thefirst term refers to the fact that the new response is re-constructed as if one receiver had recorded the responsedue to a virtual source located at the other receiver po-sition; the second indicates that the recording between

180 F. Broggini & R. Snieder

the two receivers is reconstructed through the “interfer-ence” of all the wavefields recorded at the two receiversexcited by the surrounding sources.

RARB

Scatterer

Sources

Figure 2. Green’s function reconstruction allows one to re-

construct the response between two receivers (represented by

the two triangles at locations RA and RB).

In this paper, the term focusing (Rose, 2001, 2002b)refers to the technique of finding an incident wave thatcollapses to a spatial delta function δ(x−x0) at the loca-tion x0 and at a prescribed time t0 (i.e., the wavefield isfocused at x0 at t0) as illustrated in Figure 3. In a one-dimensional medium, we deal with a one-sided problemwhen observations from only one side of the perturba-tion are available (e.g., due to the practical considera-tion that we can only record reflected waves); otherwise,we call it a two-sided problem when we have access toboth sides of the medium and account for both reflectedand transmitted waves.

t = t0t � t0

x0

Figure 3. Focusing refers to the technique of finding anincident wavefield (represented by the dashed lines) that col-

lapses to a spatial delta function δ(x−x0) at the location x0and at a prescribed time t0.

In seismology, the term imaging (Claerbout, 1985;Sava and Hill, 2009) refers to techniques that aim toreconstruct an image of the subsurface (Figure 4). Ge-ologist and geophysicists use these images to study thestructure of the interior of the Earth and to locate en-ergy resources such as oil and gas. Migration meth-ods are the most widely used imaging techniques and

their accuracy depends on the knowledge of the velocityin the subsurface. Migration methods (Bleistein et al.,2001; Biondi, 2006) involve a single scattering assump-tion (i.e., the Born approximation) because these meth-ods do not take into account the multiple reflectionsthat the waves experience during their propagation in-side the Earth; hence the data needs to be preprocessedin a specific way before such methods can be applied.

Sources

Scatterer

Receivers

Propagation Back-propagation

Figure 4. Imaging refers to techniques that aim to recon-

struct an image of the subsurface.

The ordinary form of the optical theorem (Rodbergand Thaler, 1967; Newton, 1976) relates the power ex-tinguished from a plane wave incident on a scatterer tothe scattering amplitude in the forward direction of theincident field (Figure 5). The scatterer casts a “shadow”in the forward direction where the intensity of the beamis reduced and the forward amplitude is then reducedby the amount of energy carried by the scattered wave.The generalized optical theorem (Heisenberg, 1943) isan extension of the previous theorem and it deals withthe scattering amplitude in all the directions; hence itcontains the ordinary form as a special case. This theo-rem relates the difference of two scattering amplitudesto an inner product of two other scattering amplitudes,The generalized optical theorem provides constraints onthe scattering amplitudes in many scattering problems(Marston, 2001; Carney et al., 2004). Since these theo-rems are an expression of energy conservation, they arevalid for any scattering system that does not involveattenuation (i.e., no dissipation of energy).

In this tutorial, we refer to the five subjects dis-cussed above as scattering principles because they areall related, in different ways, to a scattering process.These principles are usually studied as independentproblems but they are related in various ways; hence,understanding their connections offers insight into eachof the principles and eventually may lead to new applica-tions. This work is motivated by a simple idea: becausethe equations that rule these scattering principles havea similar functional form (see Table 1), there should bea physical connection that could lead to better compre-hension of these principles and to possible applications.

To investigate these potential connections, we fol-low two different paths to provide maximum clarity andphysical insights. We first show the relationship betweendifferent scattering principles using figures which lead

Connection of scattering principles: a visual and mathematical tour 181

Scattering loss → Scattered power

Scattering loss

Scattered power

Incident wave

Scattered power

Figure 5. The optical theorem relates the power extin-guished from a plane wave incident on a scatterer to the

scattering amplitude in the forward direction of the incident

field.

Principle Equation

1 Inverse Scattering u− u∗ =∫u∗f

2 Green’s function reconstruction G−G∗ =∫GG∗

3 Optical theorem f − f∗ =∫ff∗

4 Imaging I =∫GG∗

Table 1. Principles and their governing equations in a sim-

plified form. G is the Green’s function, f is the scatteringamplitude, and ∗ indicates complex conjugation.

the reader toward a visual understanding of the con-nections between the principles; then we illustrate andderive the mathematics that shows the link between thegoverning equations of some of these principles.

2 VISUAL TOUR

In this section, we lead the reader through a visual un-derstanding of the connections between different scat-tering principles. Figure 6 illustrates a scattering exper-iment in a one-dimensional acoustic medium where animpulsive source is placed at the position x = 1.44 kmin the model shown in Figure 8. The incident wavefield,a spatial delta function, propagates toward the discon-tinuities in the model, interacts with them, and gen-erates outgoing scattered waves. We use a time-spacefinite difference code with absorbing boundary condi-tion to simulate the propagation of the one-dimensionalwaves and to produce the numerical examples shown inthis section. The computed wavefield is shown in Fig-ure 6 and it represents the causal Green’s function ofthe system, G+. Causality ensures that the wavefield isnon zero only in the region delimited by the first arrival(i.e., the direct waves) shown in Figure 7. The slope of

the lines, representing the first arrival, depends on thevelocity model of Figure 8.

−1 0 1 2 30

1

2

3

Tim

e (

s)

Distance (km)

Figure 6. Scattering experiment with a source located at

x = 1.44 km. The traces are recorded by receivers locatedalong the model (shown in Figure 8) with a spacing of 40 m.

−1 0 1 2 30

1

2

3T

ime (

s)

Distance (km)

G+

Figure 7. Cartoon illustrating a simplified version of the

wavefield shown in Figure 6. The wavefield is non zero only

in the gray area and the black lines denote the direct waves.

Due to practical limitations, we usually are not ableto place a source inside the medium we want to probe,which raises the following question: Is it possible to cre-ate the wavefield illustrated in Figure 6 without havinga real source at the position x = 1.44 km? An initial an-swer to such a question is given by the Green’s functionreconstruction. This technique allows one to reconstructthe wavefield that propagates between a virtual sourceand other receivers located inside the medium (Wape-naar et al., 2005). We remind the reader that this tech-nique yields a combination of the causal wavefield andits time-reversed version (i.e., anti-causal). This is dueto the fact that the reconstructed wavefield is propagat-ing between a receiver and a virtual source. Without areal (physical) source, one must have non-zero incidentwaves to create waves that emanate from a receiver. Inthe next section we introduce a mathematical argumentto explain the interplay of the causal and anti-causalGreen’s functions. The fundamental steps to reconstructthe Green’s function are (Curtis et al., 2006)


−1 0 1 2 30

0.5

1

1.5

2

Ve

loc

ity

(k

m/s

)

Distance (km)

−1 0 1 2 30

0.5

1

1.5

2

De

ns

ity

(g

/cm

3)

Distance (km)

Figure 8. Velocity and density profiles of the one-

dimensional model. The perturbation in the velocity is lo-cated between x = 0.3 − 2.5 km and c0 = 1 km/s. The

perturbation in the density is located between x = 1.0 − 2.5km and ρ0 = 1 g/cm3.

1. measure the wavefields G+(x, xsl) andG+(x, xsr) at a receiver located at x (x variesfrom −1 km to 3 km) excited by impulsive sourceslocated at both sides of the perturbation xsl andxsr (a total of two sources in 1D) as shown inFigure 9;2. cross-correlate G+(x, xsl) with G+(xvs, xsl),where xvs = 1.44 km and vs stands for virtualsource;3. cross-correlate G+(x, xsr) with G+(xvs, xsr);4. sum the results computed at the two previouspoints to obtain G+(x, xvs);5. repeat this for a receiver located at a differentx.

The causal part of wavefield estimated by theGreen’s function reconstruction technique is shown inFigure 10 and it is consistent with the result of the scat-tering experiment produced with a real source locatedat x = 1.44 km, shown in Figure 6.

We thus have two different ways to reconstruct thesame wavefield, but often we are not able to access a cer-tain portion of the medium we want to study and hencewe can’t place any sources or receivers inside it. We nextassume that we only have access to scattering data R(t)measured on the left side of the perturbation, i.e. thereflected impulse response measured at x = 0 km due

−1 0 1 2 3

0

1

2

Ve

loc

ity

(k

m/s

)

Distance (km)

xsl

xsr

xvs

Figure 9. Diagram showing the locations of the real and vir-tual sources for seismic interferometry. xsl and xsr indicate

the two real sources. xvs shows the virtual source location.

−1 0 1 2 30

1

2

3

Tim

e (

s)

Distance (km)

Figure 10. Causal part of the wavefield estimated by theGreen’s function reconstruction technique when the receiver

located at x = 1.44 km acts as a virtual source.

to an impulsive source placed at x = 0 km. This furtherlimitation raises another question: Can we reconstructthe same wavefield shown in Figure 6 having knowledgeonly of the scattering data R(t)? Since there are nei-ther real sources nor receivers inside the perturbation,we speculate that the reconstructed wavefield consistsof a causal and an anti-causal part, as shown in Figure11.

For this one-dimensional problem, the answer tothis question is given by Rose (2001, 2002a). He showsthat we need a particular incident wave in order to col-lapse the wavefield to a spatial delta function at thedesired location after it interacts with the medium, andthat this incident wave consists of a spatial delta func-tion followed by the solution of the Marchenko equation,as illustrated in Figure 12.

The Marchenko integral equation (Lamb, 1980;Chadan and Sabatier, 1989) is a fundamental relationof one-dimensional inverse scattering theory. It is an in-tegral equation that relates the reflected scattering am-plitude R(t) to the incident wavefield u(t, tf ) which willcreate a focus in the interior of the medium and ulti-mately gives the perturbation of the medium. The one-dimensional form of this equation is

0 = R(t+ tf ) + u(t, tf ) +

∫ tf

−∞R(t+ t′)u(t′, tf )dt′, (1)


−1 0 1 2 3−3

−2

−1

0

1

2

3T

ime

(s

)

Distance (km)

Figure 11. Wavefield that focuses at x = 1.44 km at t = 0 s without a source or a receiver at this location. This wavefield

consists of a causal (t > 0) and an anti-causal (t < 0) part.


Deltafunction Marchenko equation

Solution of the

t

Figure 12. Incident wave that focuses at location x = 1.44

km and time t = 0 s, built using the iterative process dis-cussed by Rose (2001).

where tf is a parameter that controls the focusing lo-cation. We solve the Marchenko equation, using the it-erative process described in detail in Rose (2002a), andconstruct the particular incident wave that focuses atlocation x = 1.44 km, as shown in Figures 6 and 11.After seven steps of the iterative process, we inject theincident wave in the model from the left at x = 0 kmand compute the time-space diagram shown in the toppanel of Figure 13: it shows the evolution in time of thewavefield when the incident wave is the particular wavecomputed with the iterative method. The bottom panelof Figure 13 shows a cross-section of the wavefield atthe focusing time t = 0 s: the wavefield vanishes exceptat location x = 1.44 km; hence the wavefield focuses atthis location.

We create a focus at a location inside the pertur-bation without having a source or a receiver at sucha location and without any knowledge of the mediumproperties; we only have access to the reflected impulseresponse measured on the left side of the perturbation.With an appropriate choice of sources and receivers, thisexperiment can be done in practice (e.g., in a labora-tory).

Figure 13 however does not resemble the wavefieldshown in Figures 6 and 10. But if we denote the wave-field in Figure 13 as w(x, t) and its time-reversed ver-sion as w(x,−t), we obtain the wavefield shown in Fig-ure 11 by adding w(x, t) and w(x,−t). With this pro-cess, we effectively go from one-sided to two-sided illu-mination because in Figure 11 waves are incident on thescatterer from both sides for t < 0 s. Burridge (1980)shows similar diagrams and explains how to combinesuch diagrams using causality and symmetry proper-ties. The upper cone in Figure 11 corresponds to thecausal Green’s function and the lower cone representsthe anti-causal Green’s function; the relationship be-tween the two Green’s functions is a key element inthe next section, where we introduce the homogeneous

Green’s function Gh. Note that the wavefield in Figure11, with a focus in the interior of the medium, is basedon reflected data only. We did not use a source or re-ceiver in the medium, and did not know the medium.All necessary is encoded in the reflected waved.

Figure 14 shows four intermediate iterations of theiterative process of Rose. We see that, after addingw(x, t) to w(x,−t), the resulting wavefield is increas-ingly confined in the area defined by the causal andanti-causal cones (i.e., more energy is enclosed insidethe two cones after more iterations).

The extension of the iterative process in two di-mensions still needs to be investigated; but we concludethis section with a conjecture illustrated in Figure 15.An incident plane wave created by an array of sourcesis injected in the subsurface where it is distorted due tothe variations of the velocity inside the overburden (i.e.,the portion of the subsurface that lies above the scat-terer). Since we do not know the characteristics of thewavefield when it interacts with the region of the subsur-face that includes the scatterer, it is difficult to recon-struct the properties of the scatterer without knowingthe medium. Hence, following the insights gained withthe one-dimensional problem, we would like to createa special incident wave that, after interacting with theoverburden, collapses to a point in the subsurface creat-ing a buried source, as illustrated in Figure 16. In thiscase, assuming that the medium around the scattereris homogeneous, we would know the shape of the wave-field that probes the scatterer and partially remove theeffect of the overburden, which would facilitate accurateimaging of the scatterer.

Scatterer

Overburden

Incident plane wave

Distorted wave

x

z

Figure 15. An incident plane wave created by an array ofsources is injected in the subsurface. This plane wave is dis-

torted due to the variations in the velocity inside the over-

burden (i.e., the portion of the subsurface that lies abovethe scatterer). When the wavefield arrives in the region that

includes the scatterer, we do not know its shape.


−1 0 1 2 3−3

−2

−1

0

1

2

3T

ime

(s

)

−1 0 1 2 30

1

2

3

Am

pli

tud

e

Distance (km)

Figure 13. Top: After seven steps of the iterative process described in Rose (2002a), we inject the particular incident wave in

the model and compute the time-space diagram. Bottom: cross-section of the wavefield at t = 0 s.


−1 0 1 2 3−3

−2

−1

0

1

2

3

Tim

e (

s)

Distance (km)

Iteration 1

−1 0 1 2 3−3

−2

−1

0

1

2

3

Tim

e (

s)

Distance (km)

Iteration 2

−1 0 1 2 3−3

−2

−1

0

1

2

3

Tim

e (

s)

Distance (km)

Iteration 3

−1 0 1 2 3−3

−2

−1

0

1

2

3

Tim

e (

s)

Distance (km)

Iteration 7

Figure 14. Four different iterations of the iterative process described in Rose (2002a), we inject the particular incident wave

in the model and compute the time-space diagram by adding w(x, t) and w(x,−t). The wavefield is increasingly confined in thearea defined by the causal and anti-causal cones (i.e., more energy is enclosed inside these cones after more iterations)


Incident wave

Scatterer

Focusing

Overburden

x

z

Figure 16. Focusing of the wavefield “at depth”. A spe-cial incident wave, after interacting with the overburden, col-

lapses to a point in the subsurface creating a buried source.

3 REVIEW OF SCATTERING THEORY

We review the theory for the scattering of acousticwaves in a one dimensional medium (also called line)with a constant density, where the scatterer is repre-sented by a perturbation of a constant velocity pro-file. Here, we introduce the wave equation and Green’sfunctions that are used in the next section of the pa-per. Figure 17 shows the geometry of the scatteringproblem. The perturbation cs(x) is superposed on aconstant velocity profile c0. The following theory isdeveloped in the frequency domain because it simpli-fies the derivations (e.g., convolution becomes a mul-tiplication and derivatives become multiplications by−iω). We also show the time domain version of someof the following equations because they are more in-tuitive and allow us to understand the important roleplayed by time-reversal. The Fourier transform con-vention is defined by f(t) =

∫f(ω) exp(−iωt)dω and

f(ω) = (2π)−1∫f(t) exp(iωt)dt. Throughout this work,

when we deal with a one-dimensional problem, the di-rection of propagation n assumes only two values, 1 and−1, which correspond to waves propagating to the rightand to the left (Figure 17), respectively.

xrxA xB x

cs−1 +1

xl

c0

Figure 17. Geometry of the problem for the scattering of

acoustic waves in a one dimensional medium with constant

density. xA, xB , xl and xr are the coordinates of the receivers(represented by the triangles) and the left and right bounds of

our domain, respectively. The perturbation cs is superposed

on a constant velocity profile c0.

The equation that governs the motion of the wavesin an unperturbed medium with constant velocity c0 isthe constant-density acoustic homogeneous wave equa-

tion

L0(x, ω)u0(n, x, ω) = 0, (2)

where u0 is the displacement wavefield propagating inthe n direction and the differential operator is

L0(x, ω) ≡[d2

dx2+ω2

c20

]. (3)

The solution of equation 2 is u0(n, x, ω) =exp(inxω/c0) and its time-domain version isu0(n, x, t) = δ(t − nx/c0), which is a delta func-tion that propagates with velocity c0 in the direction nrepresenting a physical solution to the wave equation.

The unperturbed Green’s function satisfies theequation

L0(x, ω)G±0 (x, x′, ω) = −δ(x− x′), (4)

and, in the acoustic one-dimensional case, its frequency-domain expression is (Snieder, 2004):

G±0 (x, x′, ω) ≡ ± i

2ke±ik|x−x′|, (5)

where k ≡ ω/c0. The + and − superscripts of theGreen’s function represent the causal and anti-causalGreen’s function with outgoing or ingoing boundaryconditions (Oristaglio, 1989). In the time domain,causality implies that

G±0 (x, x′, t) = 0 ± t < |x− x′|/c0. (6)

Physically, the time-domain Green’s functionG±0 (x, x′, t) represents the displacement at a pointx at time t due to a point source of unit amplitudeapplied at x′ at time t = 0.

Next, we consider the interaction of the wavefieldu0 with the perturbation cs(x) (see Figure 17). Thisinteraction produces a scattered wavefield u±sc; hence,the total wavefield can be represented as u± = u0 +u±sc.The + and − superscripts in the total wavefield indicatean initial and a final condition of the wavefield in thetime domain, respectively:

u±(n, x, t)→ u0(n, x, t) t→ ∓∞. (7)

Physically, condition 7 with plus sign means that thewavefield u+, at early times, corresponds to the initialwavefield u0 propagating forward in time in the n direc-tion. The causal and anti-causal wavefields u+ and u−

are related by time-reversal; in fact, each is the time-reversed version of the other u−(t) = u+(−t). In thefrequency domain, time-reversal corresponds to complexconjugation: u−(ω) = u+∗(ω). Their time-reversal rela-tionship is better understood by comparing Figures 18aand 18b, which are valid for the velocity model of Fig-ure 8. Figure 18b is obtained by reversing the time axisof Figure 18a. We produced both figures using the samevelocity model we used in the Visual Tour section of thispaper. In Figure 18a, the initial wavefield is a narrow


−1 −0.5 0 0.5 1 1.5 2 2.50

2

4

6a

Tim

e (

s)

Distance (km)

−1 −0.5 0 0.5 1 1.5 2 2.50

2

4

6b

Tim

e (

s)

Distance (km)

Figure 18. Panel a shows the causal wavefield u+ and panel

b shows the anticausal wavefield u−. Note that each panel isthe time-reversed version of the other.

Gaussian impulse; while, in Figure 18b, the initial wave-field corresponds to the wavefield at t = 6 s in Figure18a and it coalesces to an outgoing Gaussian pulse.

The total wavefield u± satisfies the wave equation

L(x, ω)u±(n, x, ω) = 0, (8)

where the differential operator is

L(x, ω) ≡[d2

dx2+

ω2

c(x)2

]. (9)

The velocity varies with position, c(x) = c0 + cs(x),as illustrated in Figure 17. Using the operator L, wedefine the perturbed Green’s function as the functionthat satisfies the equation

L(x, ω)G±(x, x′, ω) = −δ(x− x′), (10)

with the same boundary conditions as equation 5. TheGreen’s function G± takes into account all the interac-tions with the perturbation and hence it corresponds tothe full wavefield propagating between the points x′ andx, due to an impulsive source at x′.

4 MATHEMATICAL TOUR

In this part of the paper, we lead the reader througha mathematical tour and show that the different scat-

tering principles have a common starting point, i.e., afundamental equation that reveals the connections be-tween them:

G+(xA, xB)−G−(xA, xB)

=∑

x′=xl,xr

m

[G−(x′, xA)

d

dxG+(x, xB)|x=x′

−G+(x′, xB)d

dxG−(x, xA)|x=x′

] (11)

with

m = { −1 if x′ = xl+1 if x′ = xr

,

where xA, xB , xl and xr (see Figure 17) are the coordi-nates of the receivers located at xA and xB and the leftand right bounds of our domain, respectively. Equation11 (derived in appendix A) shows a relation between thecausal and anti-causal Green’s function and we refer tothis expression, throughout the paper, as the represen-tation theorem for the homogeneous Green’s functionGh ≡ G+ − G− (Oristaglio, 1989), which satisfies thewave equation 10 when its source term is set equal tozero. G+ and G− both satisfy the same wave equation10 because the differential operator L is invariant totime-reversal (LG+ = −δ and LG− = −δ); hence, theirdifference is source free:

L(G+ −G−) = 0. (12)

The fact that G+ − G− satisfies a homogeneous equa-tion suggests that a combination of the causal and anti-causal Green’s functions is needed to focus the wavefieldat a location where there is no real source. This fact hasbeen illustrated in the previous section when we recon-structed the same wavefield using the Green’s functionreconstruction technique and the inverse scattering the-ory (see Figures 10 and 11); in both cases we obtaineda combination of the two Green’s functions.

For the remainder of this paper, to be consistentwith Budreck and Rose (1990, 1991, 1992), we use thesuperscripts + and − to indicate the causal and anti-causal behavior of wavefields and Green’s functions;and, for brevity, we omit the dependence on the angularfrequency ω.

4.1 Newton-Marchenko equation andgeneralized optical theorem

In this section, we show that equation 11 is thestarting point to derive a Newton-Marchenko equa-tion and a generalized optical theorem. In other words,we demonstrate how lines 1 and 3 of Table 1 arelinked to Gh. The Newton-Marchenko equation dif-fers from the Marchenko equation because it requiresboth reflected and transmitted waves as data (New-ton, 1980). In contrast to the Marchenko equation, theNewton-Marchenko equation can be extended to two


and three dimensions. The Marchenko and the Newton-Marchenko equations deal with the one-sided and two-sided problem, respectively. Following the work of Bu-dreck and Rose (1990, 1991, 1992), we manipulate equa-tion 11 and show how to derive the equations that rulethese two principles. Before starting with our derivation,we introduce some useful equations:

u±(n, x) = u0(n, x)

+

∫dx′G±0 (x, x′)L

′(x′)u±(n, x′),

(13)

u±(n, x) = u0(n, x)

+

∫dx′G±(x, x′)L

′(x′)u0(n, x′),

(14)

G±(n, x, x′) = G±0 (n, x, x′)

+

∫dx′′G±(x, x′′)L

′(x′′)G±0 (n, x′′, x′),

(15)

f(n, n′) = −∫dx′e−nikx′

L′(x′)u+(n′, x′), (16)

where L′(x) ≡ L(x) − L0(x) describes the influencesof the scatterer (perturbation). Equations 13, 14, and15 are three different Lippmann-Schwinger equations(Rodberg and Thaler, 1967); they are a reformulationof the scattering problem using linear integral equationswith a Green’s function kernel. The integral approachis also well suited for the study of inverse problems(Colton and Kress, 1998). Equation 16 is the scatter-ing amplitude (Rodberg and Thaler, 1967) for an inci-dent wave traveling in the n direction and that is scat-tered in the n′ direction. We insert equation 15 into 11,simplify considering the fact that xr > xA, x

′′, xB andxl < xA, x

′′, xB , and using expression 14:

G+(xA, xB)−G−(xB ,xA)

=

(i

2k

)(− i

2k

)[ iku−(+1, xA)u+(−1, xB)

+iku+(−1, xB)u−(+1, xA)

+iku−(−1, xA)u+(+1, xB)

+iku+(+1, xB)u−(−1, xA) ] .

(17)

In a more compact form this can be written as

G+(xA,xB)−G−(xB , xA)

=

(i

2k

) ∑n=−1,1

u−(n, xA)u+(−n, xB).(18)

Equation 18 is the starting point to derive a Newton-Marchenko equation; we show the full derivation in Ap-

pendix B and write the final result:

u+(+1, xA)− u−(+1, xA)

= − i

2k

∑n=−1,1

u−(n, xA)f(n, xB), xB > xA, x′′,

(19)

and

u+(−1, xA)− u−(−1, xA)

= − i

2k

∑n=−1,1

u−(n, xA)f(n,−xB), xB < xA, x′′.

(20)

The system of coupled equations 19 and 20 is our rep-resentation of the one-dimensional Newton-Marchenkoequation; recognizing that u− = u∗, it corresponds toline 1 of Table 1.

Next, from expression 19 and using equation 13,u−(n, x) = u+∗(−n, x), and u± = u0 + u±sc, we obtain ageneralized optical theorem:

<f(−n, n) = −∑

n′=−1,1

f(−n, n′)f∗(n, n′), (21)

where n assumes the value −1 or +1, and < indicatesthe real part (see line 3 of Table 1). The obtained resultsare exact because in this one-dimensional framework wedo not use any far-field approximations.

4.2 Green’s function reconstruction and theOptical theorem

Starting from the three-dimensional version of equation11, Snieder et al. (2008) showed the connection betweenthe generalized optical theorem and the Green’s func-tion reconstruction. Following their three-dimensionalformulation, we illustrate the same result for the one-dimensional problem and show the connection betweenlines 2 and 3 of Table 1. In this part of the paper, weemphasize the physical meaning of this connection andshow the mathematical derivation in the Appendix C.The expression for the Green’s function reconstructionis

i

2k[ G+(xA,xB)−G−(xA, xB) ]

=∑

x′=xl,xr

G+(xA, x′)G−(xB , x

′).(22)

Unlike the three-dimensional case, we obtain two differ-ent results depending on the configuration of the sys-tem. In the first case (Figure 19a), the ordinary opticaltheorem is derived:

<f(n, n) = −∑

n′=−1,1

|f(n, n′)|2; (23)

in the second case (Figure 19b), we obtain a generalizedoptical theorem

<f(−n, n) = −∑

n′=−1,1

f(−n, n′)f∗(n, n′), (24)


where n assumes the value −1 or +1 and < indicatesthe real part in both cases.

xrxA xB xxl 0

0

cs

cs

a

b

c0

c0

xA xB

c0tA c0tB

Figure 19. Configurations of the system used to show the

connection between the Green’s function reconstruction and

the optical theorem. In both cases, the receivers xA and xBare located outside the scatterer cs, which is located at x = 0.

The above expressions of the optical theorem in onedimension agree with the work of Hovakimian (2005)and differ from their three-dimensional counterpart be-cause they contain the real part of the scattering ampli-tude instead of the imaginary part. The connection be-tween the Green’s function reconstruction and the gen-eralized optical theorem has not only a mathematicalproof, but a physical explanation. The cross-correlationof scattered waves in equation 22 produces a spuriousarrival (Snieder et al., 2008), i.e. an unphysical wavethat is not predicted by the theory. In the first config-uration shown in Figure 19a, such spurious arrival hasthe same arrival time as the direct wave, tB + tA, butits amplitude is not correct (see term T3 in equationC3). In the second case (Figure 19b), tA and tB corre-spond to the time that a wave takes to travel from theorigin x = 0 to xA and xB , respectively. Here, the spu-rious arrival corresponds to a wave that arrives at timetB − tA when no physical wave arrives; in fact it wouldarrive before the direct arrival at time tB+tA. But, sincethe ordinary and generalized optical theorem hold, thespurious arrival cancels in both cases (as shown in Ap-pendix C). We note that the ordinary form of the opticaltheorem 23 also follows from its generalized form 24 (theformer is a special case of the latter); and that equation24 is equivalent to the expression for the optical theoremderived in the previous section, equation 21.

4.3 Green’s function reconstruction andImaging in 3D

The three-dimensional version of equation 11 is givenby:

G+(rA, rB)−G−(rA, rB)

=

∮∂V

1

ρ[G−(r, rA)∇G+(r, rB)

−G+(r, rB)∇G−(r, rA) ]n · d2r.

(25)

Equation 25 is derived from an acoustic reciprocitytheorem of the cross-correlation type, the equation ofmotion, and the constitutive relation (Wapenaar andFokkema, 2006). A factor of ρ−1 appears in 25 becauseWapenaar and Fokkema (2006) define the Green’s func-tion for a wave equation with a forcing term −ρδ(r−r′).

V

∂V

Scatterer

n

Figure 20. Geometry of the scattering problem in three spa-

tial dimensions. The domain consists of a volume V , boundedby a surface ∂V . The unit vector normal to ∂V is represented

by n.

If the sources are in the far-field with respect to thereceivers and if the surface ∂V is a sphere with a largeradius, we can apply a radiation boundary condition(Bleistein et al., 2001) to the Green’s function:

∇G±(rA,B , r) = ±ikG±(rA,B , r)n, (26)

where n is a unit vector normal to the surface ∂V , seeFigure 20. Applying this boundary condition to equa-tion 25, we obtain a simplified expression in the fre-quency domain for the Green’s function reconstruction:

G+(rA, rB)−G−(rA, rB)

= 2iω

∮∂V

1

ρcG−(r, rA)G+(r, rB)d2r.

(27)

Equation 27 corresponds to line 2 of Table 1. In globaland exploration seismology this is the equation foracoustic seismic interferometry or the virtual sourcemethod.

We next establish the connection between equation25 and imaging. Oristaglio (1989) gives an unusual rep-resentation of the spatial delta function that is similar to


the right-hand side of equation 25. This representationcorresponds to the time derivative evaluated at t = 0 ofthe time-domain version of Gh(ω):

δ(r− r′) =1

2πc2

∫ +∞

−∞dω(−iω)

×∮∂V

[ G−(r, r′′, ω)∇G+(r′′, r′, ω)

−G+(r′′, r′, ω)∇G−(r, r′′, ω) ]n · d2r′′.

(28)

According to equation 25, the expression inside theparentheses is the homogeneous Green’s function Gh =G+ −G− (except for a factor ρ−1). Applying the radi-ation boundary condition 26 to equation 28 and settingr = r′, gives:

δ(r− r′) =1

π

∮∂V

d2r′′

×∫ +∞

−∞

ω2

c3G−(r, r′′, ω)G+(r′′, r′, ω)dω.

(29)

When r → r′, this representation of the spatial deltafunction can be thought of as an imaging condition∫ +∞−∞ dω G−R(ω)G+

S (ω) (Claerbout, 1985), which is theprocess used in different migration techniques to pro-duce an image, after propagating the source arrayG+

S (ω) and back-propagating the receiver array G−R(ω)(see line 4 of Table 1). The main idea behind an imag-ing condition is that the two wavefields are kinemati-cally equivalent only at the reflector positions, hence thecross-correlation of such wavefields yields an image ofthe subsurface. The causal and anticausal Green’s func-tions represent the operators that propagate the sourceand receiver wavefields in the subsurface; their interac-tion inside equation 29 eventually produces an image.Furthermore, equation 29 can be interpreted as an ex-tended imaging condition, where (r− r′) represents thespace-lag extension (Rickett and Sava, 2002; Sava andVasconcelos, 2010).

5 CONCLUSIONS

In the Visual Tour section, we described the connectionbetween different scattering principles, showing thatthere are three distinct ways to reconstruct the samewavefield. A physical source, the Green’s function recon-struction technique, and inverse scattering theory allowone to create the same wavestate (see Figure 6) origi-nated by an impulsive source placed at a certain loca-tion xs (x = 1.44 km in our examples). Green’s func-tion reconstruction tells us how to build an estimate ofthe wavefield without knowing the medium properties,if we have a receiver at the same location xs of the realsource and sources surrounding the scattering region.Inverse scattering goes beyond and allows us to focusthe wavefield inside the medium (at location xs) with-out knowing its properties, using only data recorded at

one side of the medium. We showed that the interac-tion between causal and anti-causal wavefields is a keyelement to focus the wavefield where there is no realsource; in fact Gh = G+ − G−, which satisfies the ho-mogeneous wave equation 10, is a superposition of thecausal and anti-causal Green’s functions.

We speculate that many of the insights gained inour one-dimensional framework are still valid in higherdimensions. An extension of this work in two or threedimensions would give us the theoretical tools for manyuseful practical applications. For example, if we knewhow to create the three-dimensional version of the in-cident wavefield shown in Figure 12, we could focusthe wavefield to a point in the subsurface to simulatea source at depth and to record data at the surface(Figure 16); these kind of data are of extreme impor-tance for full waveform inversion techniques (Brendersand Pratt, 2007) and subsalt imaging (Sava and Biondi,2004). Furthermore, we could possibly concentrate theenergy of the wavefield inside a hydrocarbon reservoirto fracture the rocks and improve the production of oiland gas (Beresnev and Johnson, 1994).

In the second part of this tutorial, the Mathemati-cal Tour, we demonstrated that the representation the-orem for the homogeneous Green’s function Gh , equa-tion 11, constitutes a theoretical framework for vari-ous scattering principles. We showed that all the prin-ciples and their equations (see Table 1) rely on Gh as astarting point for their derivation. As mentioned above,the fundamental role played by the combination of thecausal and anti-causal Green’s functions has been evi-dent throughout all the Mathematical Tour: it is thiscombination that allows one to focus the wavefield to alocation where neither a real source nor a receiver canbe placed.

ACKNOWLEDGMENTS

The authors would like to thank the members of theCenter for Wave Phenomena for their constructive com-ments. This work was supported by the sponsors ofthe Consortium Project on Seismic Inverse Methods forComplex Structures at the Center for Wave Phenomena.

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APPENDIX A: DERIVATION OF EQUA-TION 11

Here, we derive equation 11, i.e., the representation the-orem for the homogeneous Green’s function. We start


with the equations

d2

dx2G+(x, xB) +

ω2

c(x)2G+(x, xB) = −δ(x− xB) (A1)

and

d2

dx2G−(x, xA) +

ω2

c(x)2G−(x, xA) = −δ(x− xA), (A2)

where xA, xB , and x indicate a position between xland xr in Figure 17. Next, we multiply equation A1by G−(x, xA), and equation A2 by G+(x, xB); then wesubtract the two results and integrate between xl andxr, yielding

G+(xA, xB)−G−(xB , xA)

=

∫ xr

xl

dx

[G−(x, xA)

d2

dx2G+(x, xB)

−G+(x, xB)d2

dx2G−(x, xA)

].

(A3)

The right-hand side of the last equation is an exactderivative: ∫ xr

xl

dx

[G−(x, xA)

d2

dx2G+(x, xB)

−G+(x, xB)d2

dx2G−(x, xA)

]≡∫ xr

xl

dxd

dx

[G−(x, xA)

d

dxG+(x, xB)

−G+(x, xB)d

dxG−(x, xA)

];

(A4)

hence we obtain the expression for Gh:

G+(xA, xB)−G−(xA, xB)

=∑

x′=xl,xr

m

[G−(x′, xA)

d

dxG+(x, xB)|x=x′

−G+(x′, xB)d

dxG−(x, xA)|x=x′

](A5)

with

m = { −1 if x′ = xl+1 if x′ = xr

,

where we have used the source-receiver reciprocity rela-tion G±(xA, xB) = G±(xB , xA) for the acoustic Green’sfunction (Wapenaar and Fokkema, 2006).

APPENDIX B: DERIVATION OF THENEWTON-MARCHENKO EQUATION

Inserting expession 15 into the left-hand side of equation18, using the relation u+ = u0 + u+

sc in the right-handside of 18, and then inserting 13 into the right-hand side,

we get

e+ik|xA−xB | +

∫dx′′G+(xA, x

′′)L′(x′′)e+ik|x′′−xB |

+e−ik|xA−xB | +

∫dx′′G−(xA, x

′′)L′(x′′)e−ik|x′′−xB |

= u−(+n,xA)e−ikxB + u+(+n, xA)e+ikxB

+i

2ku−(+n, xA)

×∫dx′′e+ik|xB−x′′|L

′(x′′)u+(−n, x′′)

+i

2ku−(−n, xA)

×∫dx′′e+ik|xB−x′′|L

′(x′′)u+(+n, x′′).

(B1)

In this one-dimensional problem, we need to considertwo different cases: 1) xB > xA, x

′′ and 2) xB < xA, x′′.

Without loss of generality we choose xB > xA, x′′, and

hence obtain

e+ikxB

[e−ikxA +

∫dx′′G+(xA, x

′′)L′(x′′)e−ikx′′

]+e−ikxB

[eikxA +

∫dx′′G−(xA, x

′′)L′(x′′)e+ikx′′

]= u−(+1,xA)e−ikxB + u−(−1, xA)e+ikxB

− i

2ku−(+1, xA)e+ikxB

×∫−dx′′e−ikx′′

L′(x′′)u+(−1, x′′)

− i

2ku−(−1, xA)e+ikxB

×∫−dx′′e−ikx′′

L′(x′′)u+(+1, x′′).

(B2)

The terms inside the brackets in the left-hand side cor-respond to u+(−1, xA) and u−(+1, xA), respectively;while the integrals in the right-hand side correspondto f(+1,−1) and f(+1,+1), respectively. We rewriteequation B2 using 13, 14, and the relation f(+n,+n′) =f(−n′,−n), to give

u+(+1, xA)− u−(+1, xA) =

− i

2k

∑n=−1,1

u−(n, xA)f(n, xB).(B3)

For the second case xB < xA, x′′, the solution is

u+(−1, xA)− u−(−1, xA) =

− i

2k

∑n=−1,1

u−(n, xA)f(n,−xB).(B4)


APPENDIX C: GREEN’S FUNCTION RECONSTRUCTION AND THE OPTICAL THEOREM

In this appendix we derive the mathematics that shows the connection between the Green’s function reconstructionequation and the optical theorem in one dimension. The expression for the Green’s function reconstruction is

i

2k

[G+(xA, xB)−G−(xA, xB)

]=

∑x′=xl,xr


′), (C1)

and the Green’s function excited by a point source at xs recorded at xr is given by

G+(xr, xs) = − i

2keik|xs−xr|︸︷︷︸Td

− i

2keik|xs|f(n, n′)eik|xr|︸︷︷︸

Ts

, (C2)

where f(n, n′) represents the scattering amplitude (Rodberg and Thaler, 1967), and n′ and n represent the directionsof the incident wave and the scattered wave, respectively. In the expression above, Td represents the wave travelingdirectly from the source to the receiver, and term Ts corresponds to the scattered wave that reaches the receiversafter interacting with the scatterer. Considering the first configuration (Figure 19a), inserting equation C2 into theright-hand side of equation C1 we get∑

x′=xl,xr


′) =

− i

2k

[− i

2keik(xB−xA) − i

2ke−ik(xB+xA)f(−1, 1)

]︸︷︷︸

T1

− i

2k

[− i

2ke−ik(xB−xA) − i

2keik(xB+xA)f∗(−1, 1)

]︸︷︷︸

T2

−(i

2k

)(i

2k

)eik(xB−xA) [ f(−1,−1) + f∗(−1,−1) + |f(−1,−1)|2 + |f(−1, 1)|2 ]︸︷︷︸

T3

.

(C3)

The terms T1 and T2 correspond to G+(xA, xB) and −G−(xA, xB), respectively, while the term T3 represents theunphysical wave previously discussed in the Mathematical tour; hence, equation C3 simplifies to∑

x′=xl,xr


′)

=i

2k

[G(xA, xB)−G−(xA, xB)

]−(i

2k

)2

eik(xB−xA) [ f(−1,−1) + f∗(−1,−1) + |f(−1,−1)|2 + |f(−1, 1)|2 ] .

(C4)

For the right-end side of equation C4 to be equal to the left-hand side of equation C1, the expression between thesquare brackets in term T3 should vanish:

f(−1,−1) + f∗(−1,−1) = −|f(−1,−1)|2 + |f(−1, 1)|2. (C5)

Equation C5 is the expression for the one-dimensional optical theorem (Hovakimian, 2005). The second configuration(Figure 19b) gives∑

x′=xl,xr


′)

=i

2k

[G+(xA, xB)−G−(xA, xB)

]−(i

2k

)2

eik(xB+xA) [ f(−1, 1) + f∗(1,−1) + f(−1,−1)f∗(1,−1) + f(−1, 1)f∗(1, 1) ]︸︷︷︸T4

.

(C6)

In this case, term T4 corresponds to the generalized optical theorem in one dimension (Hovakimian, 2005).

connection of scattering principles: a visual and

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