Seismic interferometry by crosscorrelation and by multi ... · Seismic interferometry, also known...

CWP-689

Seismic interferometry by crosscorrelation and bymulti-dimensional deconvolution: a systematiccomparison

Kees Wapenaar1, Joost van der Neut1, Elmer Ruigrok1, Deyan Draganov1,

Jurg Hunziker1, Evert Slob1, Jan Thorbecke1, Roel Snieder2

(1) Department of Geotechnology, Delft University of Technology, P.O. Box 5048, 2600 GA Delft, The Netherlands,

(2) Center for Wave Phenomena, Colorado School of Mines, Golden, Colorado 80401, USA.

ABSTRACTSeismic interferometry, also known as Green’s function retrieval by crosscorrelation, has a

wide range of applications, ranging from surface-wave tomography using ambient noise, to

creating virtual sources for improved reflection seismology. Despite its successful applications,

the crosscorrelation approach also has its limitations. The main underlying assumptions are

that the medium is lossless and that the wave field is equipartitioned. These assumptions

are in practice often violated: the medium of interest is often illuminated from one side only,

the sources may be irregularly distributed, and losses may be significant. These limitations

may partly be overcome by reformulating seismic interferometry as a multidimensional de-

convolution (MDD) process. We present a systematic analysis of seismic interferometry by

crosscorrelation and by MDD. We show that for the non-ideal situations mentioned above,

the correlation function is proportional to a Green’s function with a blurred source. The

source blurring is quantified by a so-called interferometric point-spread function which, like

the correlation function, can be derived from the observed data (i.e., without the need to

know the sources and the medium). The source of the Green’s function obtained by the corre-

lation method can be deblurred by deconvolving the correlation function for the point-spread

function. This is the essence of seismic interferometry by MDD. We illustrate the crosscorre-

lation and MDD methods for controlled-source and passive-data applications with numerical

examples and discuss the advantages and limitations of both methods.

Key words: seismic interferometry, crosscorrelation, deconvolution, ambient noise, virtual

source

1 INTRODUCTION

In recent years, the methodology of Green’s functionretrieval by crosscorrelation has led to many applica-tions in seismology (exploration, regional and global),underwater acoustics and ultrasonics. In seismology thismethodology is also called “seismic interferometry”.One of the explanations for the broad interest lies in thefact that new responses can be obtained from measureddata in a very simple way. In passive-data applications,a straightforward crosscorrelation of responses at tworeceivers gives the impulse response (Green’s function)at one receiver of a virtual source at the position of the

other. In controlled-source applications the procedure issimilar, except that it involves in addition a summationover the sources. For a review of the theory and themany applications of seismic interferometry, see Laroseet al. (2006), Schuster (2009) and Snieder et al. (2009).

It has also been recognized that the simple crosscor-relation approach has its limitations. From the varioustheoretical models it follows that there is a number ofunderlying assumptions for retrieving the Green’s func-tion by crosscorrelation. The most important assump-tions are that the medium is lossless and that the wavesare equipartitioned. In heuristic terms, the latter con-dition means that the receivers are illuminated isotrop-

152 Wapenaar et al.

ically from all directions, which is for example achievedwhen the sources are regularly distributed along a closedsurface, the sources are mutually uncorrelated and theirpower spectra are identical. Despite the fact that inpractical situations these conditions are at most onlypartly fulfilled, the results of seismic interferometry aregenerally quite robust, but the retrieved amplitudes areunreliable and the results are often blurred by artifacts.

Several researchers have proposed to address someof the shortcomings by replacing the correlation pro-cess by deconvolution. In the “virtual source method”,pioneered by Bakulin & Calvert (2004), the main lim-itation is that the real sources are present only at theEarth’s surface, i.e., above the downhole receivers, in-stead of on a closed surface surrounding the receivers, asprescribed by the theory (Wapenaar et al. 2005; van Ma-nen et al. 2005; Korneev & Bakulin 2006). To compen-sate for this one-sided illumination, Bakulin & Calvert(2006) propose to replace the correlation by a decon-volution for the downgoing wave field at the downholereceivers. They show that this approach compensatesfor variations in the source wavelet and, partly, for re-verberations in the overburden. Snieder et al. (2006a)deconvolve passive wave fields observed at differentdepth levels and show that, apart from compensatingfor the source function, this methodology also changesthe boundary conditions of the system. They applied itto earthquake data recorded at different heights in theMillikan library in Pasadena and obtained the impulseresponse of the building. Mehta et al. (2007b) used asimilar approach to estimate the near-surface proper-ties of the Earth from passive recordings in a verticalborehole. Vasconcelos & Snieder (2008a,b) and Vascon-celos et al. (2008) used deconvolution interferometry inseismic imaging with complicated and unknown source-time signals and for imaging with internal multiples.All these approaches improve seismic interferometry tosome extent. An important positive aspect is that therequired processing in all these cases is not much morecomplicated than in the correlation approach becausethe employed deconvolution procedures are essentially1-D (i.e., trace-by-trace deconvolution). This compen-sates for variations in the source wavelet, anelastic lossesand, partly, for internal multiples, but it does not ac-count for the anisotropic illumination of the receivers.

To obtain more accurate results, seismic interferom-etry by deconvolution should acknowledge the 3-D na-ture of the seismic wave field. Hence, from a theoreticalpoint of view, the trace-by-trace deconvolution processshould be replaced by a full 3-D wave field deconvolutionprocess. In the following we speak of multi-dimensionaldeconvolution (MDD), which stands for 3-D wave fielddeconvolution in the 3-D situation, or for 2-D wave fielddeconvolution in the 2-D situation. The MDD princi-ple is not entirely new. It has been applied for examplefor multiple elimination of ocean-bottom data (Wape-naar et al. 2000; Amundsen 2001). Like the aforemen-

tioned method of Snieder et al. (2006a), this can beseen as a methodology that changes the boundary con-ditions of the system: it transforms the response of thesubsurface including the effects of the reflecting oceanbottom and water surface into the response of a subsur-face without these reflecting boundaries. With hindsightthis methodology appears to be an extension of a 1-Ddeconvolution approach proposed by Riley & Claerbout(1976). Muijs et al. (2007) employ multi-dimensionaldeconvolution for the downgoing wave field (includingmultiples) in the imaging condition, thus improving theillumination and resolution of the imaged structures.Schuster & Zhou (2006) and Wapenaar et al. (2008a)discuss MDD of controlled-source data in the contextof seismic interferometry. Slob et al. (2007) apply MDDto modeled controlled-source electromagnetic (CSEM)data and demonstrate the relative insensitivity to dissi-pation as well as the potential of changing the boundaryconditions: the effect of the air wave, a notorious prob-lem in CSEM prospecting, is largely suppressed. Wape-naar et al. (2008b) apply MDD to passive seismic dataand show how it corrects for anisotropic illuminationdue to an irregular source distribution. Van der Neut &Bakulin (2009) show how MDD can be used to improvethe radiation pattern of a virtual source. In a relatedmethod, called “directional balancing”, Curtis & Halli-day (2010) deconvolve the crosscorrelation result by theestimated radiation pattern of the virtual source.

Interferometry by MDD is more accurate than thetrace-by-trace correlation and deconvolution approachesbut the processing is more involved. In this paper wepresent a systematic analysis of seismic interferometryby crosscorrelation versus multi-dimensional deconvolu-tion and discuss applications of both approaches.

2 GREEN’S FUNCTIONREPRESENTATIONS FOR SEISMICINTERFEROMETRY

We briefly review two Green’s function representationsfor seismic interferometry. Consider a volume V enclosedby a surface S, with outward pointing normal vectorn = (n1, n2, n3). In V we have an arbitrary inhomoge-neous medium with acoustic propagation velocity c(x)and mass density ρ(x) (where x = (x1, x2, x3) is theCartesian coordinate vector). We consider two points inV, denoted by coordinate vectors xA and xB , see Figure1a. We define the Fourier transform of a time-dependentfunction u(t) as u(ω) =

R ∞−∞ exp(−jωt)u(t)dt, with j

the imaginary unit and ω the angular frequency. As-suming the medium in V is lossless, the correlation-typerepresentation for the acoustic Green’s function betweenxA and xB in V reads (Wapenaar et al. 2005; van Manen

A comparison of approaches in seismic interferometry 153

Axˆ ( , , )B AG !x x

ˆ ( , , )BG !x x

ˆ ( , , )AG !x x

a)

n

x

Bx

VS

Sx

Bx

ˆ ( , , )B SG !x x

ˆ ( , , )BG !x x

x

b)ˆ ( , , )SG !x x

Sn

V

Figure 1. (a) Configuration for the correlation-type Green’s

function representation (equation 1). The medium in V is as-

sumed lossless. The rays represent full responses, includingprimary and multiple scattering due to inhomogeneities in-

side as well as outside S. The bar in ˆG refers to a referencestate with possibly different boundary conditions at S and/or

different medium parameters outside S (in V the medium pa-

rameters are the same for G and ˆG). (b) Configuration forthe convolution-type Green’s function representation (equa-

tion 2). Here the medium does not need to be lossless.

et al. 2005)

ˆG(xB ,xA, ω) + G∗(xB ,xA, ω) =

−I

S

1

jωρx)

“∂i

ˆG(xB ,x, ω)G∗(xA,x, ω) (1)

− ˆG(xB ,x, ω)∂iG∗(xA,x, ω)

”ni dx

(Einstein’s summation convention applies to repeatedlower case Latin subscripts). This representation is abasic expression for seismic interferometry (or Green’sfunction retrieval) by crosscorrelation in open systems.The superscript asterisk ∗ denotes complex conjugation,

hence, the products on the right-hand side correspondto crosscorrelations in the time domain of observationsat two receivers at xA and xB . The notation ˆG is intro-duced to denote a reference state with possibly differentboundary conditions at S and/or different medium pa-rameters outside S (but in V the medium parameters

for ˆG are the same as those for G). The bar is usu-

ally omitted because G and ˆG are usually defined inthe same medium throughout space. The Green’s func-tions on the left-hand side are the Fourier transformsof the response of a source at xA observed at xB andits time-reversed version. Representation (1) is exact,hence, it accounts not only for the direct wave, but alsofor primary and multiply scattered waves. Note thatthe inverse Fourier transform of the left-hand side givesG(xB ,xA, t) + G(xB ,xA,−t), from which G(xB ,xA, t)and G(xB ,xA,−t) can be obtained by extracting thecausal and acausal part, respectively. Porter (1970), Es-mersoy & Oristaglio (1988) and Oristaglio (1989) usedan expression similar to equation (1) (without the bars)in optical holography, seismic migration and acousticinverse scattering, respectively, see Thorbecke & Wape-naar (2007) and Halliday & Curtis (2010) for a furtherdiscussion on the relation between these different appli-cations of equation (1).

Next, we consider a convolution-type represen-tation for the Green’s function. We slightly modifythe configuration by taking xA outside S and renam-ing it xS , see Figure 1b. For this configuration theconvolution-type representation is given by

G(xB ,xS , ω) =

−I

S

1

jωρ(x)

“∂i

ˆG(xB ,x, ω)G(x,xS , ω) (2)

− ˆG(xB ,x, ω)∂iG(x,xS , ω)”ni dx.

Due to the absence of complex-conjugation signs, theproducts on the right-hand side correspond to crosscon-volutions in the time domain. An important differencewith the correlation-type representation is that this rep-resentation remains valid in media with losses. Again,

the bar in ˆG refers to a reference state with possi-bly different boundary conditions at S and/or differentmedium parameters outside S. Slob & Wapenaar (2007)use the electromagnetic equivalent of equation (2) (with-out the bars) as the starting point for interferometry bycrossconvolution in lossy media. A discussion of interfer-ometry by crossconvolution is beyond the scope of thispaper. We will use equation (2) as the starting point forinterferometry by deconvolution in open systems. By

considering ˆG(xB ,x, ω) under the integral as the un-known quantity, equation (2) needs to be resolved bymulti-dimensional deconvolution.

154 Wapenaar et al.

3 BASIC ASPECTS OFINTERFEROMETRY BYCROSSCORRELATION

3.1 Simplification of the integral

The correlation-type Green’s function representation(equation 1) is a basic expression for seismic interfer-ometry by crosscorrelation in open systems. The right-hand side of this equation contains a combination oftwo correlation products. We show how we can com-bine these two terms into a single term. To this end, weassume that the medium at and outside S is homoge-neous, with constant propagation velocity c and mass

density ρ, for G as well as for ˆG. In other words, S is an

absorbing boundary for G and ˆG. Because the bound-

ary conditions are the same, we drop the bar in ˆG inthe remainder of this section. Assuming S is sufficientlysmooth, the normal differential operator ni∂i acting onthe Green’s functions can be replaced by a pseudo-differential operator −jH1, where H1 is the square-rootof the Helmholtz operator defined on curvilinear coor-dinates along S. For details about this operator we referto Fishman & McCoy (1984), Wapenaar & Berkhout(1989, Appendix B), Fishman (1993), and Frijlink &Wapenaar (2010). Hence, for the integral in equation(1) we may writeI

S

“(∂iGB)G∗

A − GB(∂iG∗A)

”ni dx

=

IS

“(−jH1GB)G∗

A − GB(jH∗1G∗

A)”

dx (3)

=

IS

“−j(H1 + H∗

1)GB

”G∗A dx,

where we used the fact that H1 is symmetric [in thesense that

H(H1f)g dx =

Hf(H1g) dx]. Note that GA

and GB stand for G(xA,x, ω) and G(xB ,x, ω), respec-tively. Zheng (2010) and Zheng et al. (2011) evaluatethe kernel of −j(H1 + H∗

1) analytically for a number ofspecial cases and show that in theory the last integralin equation (3) may lead to an exact retrieval of theGreen’s function. For most practical applications somefurther approximations need to be made. If we ignorethe contribution of evanescent waves, we may approxi-mate H∗

1 by H1, henceIS

“−j(H1 + H∗

1)GB

”G∗A dx ≈

2

IS(−jH1GB)G∗

A dx = 2

IS(ni∂iGB)G∗

A dx. (4)

Using equations (3) and (4) in equation (1) andrewriting x as xS (standing for the source coordinatevector), we obtain

G(xB ,xA, ω) + G∗(xB ,xA, ω) = (5)

− 2

jωρ

ISsrc

`ni∂

Si G(xB ,xS , ω)

´G∗(xA,xS , ω) dxS ,

where the superscript S in ∂Si denotes that the differ-entiation is carried out with respect to the componentsof xS . Note that we added a subscript ‘src’ in Ssrc todenote that the integration surface contains the sourcesof the Green’s functions. The integrand of equation (5)contains a single crosscorrelation product of dipole andmonopole source responses. When only monopole re-sponses are available, the operation ni∂

Si can be re-

placed by the pseudo-differential operator −jH1 actingalong Ssrc, or in the high-frequency approximation bymultiplications with −jk | cos α|, where k = ω/c, and αis the angle between the relevant ray and the normal onS. Hence, for controlled-source interferometry, in whichcase the source positions are known and Ssrc is usuallya smooth surface, equation (5) is a useful expression. Inpassive interferometry, the positions of the sources areunknown and Ssrc can be very irregular. In that casethe best one can do is replace the operation ni∂

Si by a

factor −jk, which leads to

G(xB ,xA, ω) + G∗(xB ,xA, ω) ≈ (6)

2

ρc

ISsrc

G(xB ,xS , ω)G∗(xA,xS , ω) dxS .

Equation (6) is accurate when Ssrc is a sphere with avery large radius, but it involves amplitude errors whenSsrc is finite. Moreover, spurious events may occur dueto incomplete cancelation of contributions from differ-ent stationary points. However, since the approximationdoes not affect the phase, equation (6) is usually consid-ered acceptable for seismic interferometry. Note that fora modified Green’s function G = G/jω we have, insteadof equation (6),

G(xB ,xA, ω)− G∗(xB ,xA, ω) ≈ (7)

−2jω

ρc

ISsrc

G(xB ,xS , ω)G∗(xA,xS , ω) dxS

(Wapenaar & Fokkema 2006). Both equations (6) and(7) are used in the literature on seismic interferometry.Because of the simple relation between G and G, theseequations are completely interchangeable. In the follow-ing we continue with equation (6). In the time domainthis equation reads

G(xB ,xA, t) + G(xB ,xA,−t) ≈ (8)

2

ρc

ISsrc

G(xB ,xS , t) ∗G(xA,xS ,−t) dxS .

The in-line asterisk ∗ denotes temporal convolution, butthe time-reversal of the second Green’s function turnsthe convolution into a correlation. Equation (8) showsthat the crosscorrelation of two Green’s functions ob-served by receivers at xA and xB , followed by an in-tegration along the sources, gives the Green’s functionbetween xA and xB plus its time-reversed version.


Bx

srcS

( , , )B Su tx x( , , )A Su tx x

( , , ) ( )B AG t S t!x x!

a)

AxV

Sx""" """

( , , )A Su tx x

( , , )B Su tx x( , , ) ( )B AG t S t!x x!

Sx""" """

Bx

srcS

Ax

b)

V

Figure 2. (a) Illustration of interferometry by crosscorrela-

tion (equations (11) and (13)) for the situation of direct-wave

interferometry. The responses u(xA,xS , t) and u(xB ,xS , t)are crosscorrelated and the results for all sources xS are inte-

grated along the source boundary Ssrc. The correlation func-

tion is approximately proportional to G(xB ,xA, t)∗S(t). (b)Idem, for reflected-wave interferometry.

3.2 Transient and noise sources

For practical situations the Green’s functions in theright-hand side of equation (8) should be replaced byresponses of real sources, i.e., Green’s functions con-volved with source functions. When the source functionsare transients, s(xS , t), we write for the responses at xAand xB

u(xA,xS , t) = G(xA,xS , t) ∗ s(xS , t), (9)

u(xB ,xS , t) = G(xB ,xS , t) ∗ s(xS , t). (10)

For this situation, we define a correlation functionC(xB ,xA, t) as

C(xB ,xA, t) = (11)ISsrc

F(xS , t) ∗ u(xB ,xS , t) ∗ u(xA,xS ,−t) dxS ,

where F(xS , t) is a filter that compensates for the vari-ations of the autocorrelation of the source function, insuch a way that

F(xS , t) ∗ s(xS , t) ∗ s(xS ,−t) = S(t), (12)

where S(t) is some (arbitrarily chosen) average autocor-relation. Substituting equations (9) and (10) into equa-tion (11), using equation (12), and comparing the resultwith equation (8) gives

{G(xB ,xA, t)+G(xB ,xA,−t)}∗S(t) ≈ 2

ρcC(xB ,xA, t).

(13)

When the source functions are simultaneously act-ing noise signals, N(xS , t), the responses at xA and xBare given by

u(xA, t) =

ISsrc

G(xA,xS , t) ∗N(xS , t) dxS , (14)

u(xB , t) =

ISsrc

G(xB ,x′S , t) ∗N(x′S , t) dx′S . (15)

We assume that two noise sources N(xS , t) and N(x′S , t)are mutually uncorrelated for any xS 6= x′S on Ssrc

and that their autocorrelations are independent of xS .Hence, we assume that these noise sources obey the re-lation

〈N(x′S , t) ∗N(xS ,−t)〉 = δ(xS − x′S)S(t), (16)

where 〈·〉 denotes ensemble averaging and S(t) the auto-correlation of the noise. Note that δ(xS −x′S) is definedfor xS and x′S both on Ssrc. This time we define thecorrelation function as

C(xB ,xA, t) = 〈u(xB , t) ∗ u(xA,−t)〉. (17)

In practice, the ensemble averaging is replaced by inte-grating over sufficiently long time and/or averaging overdifferent time intervals. Substituting equations (14) and(15) into equation (17), using equation (16), and com-paring the result with equation (8) gives again equa-tion (13). Hence, whether we consider transient or noisesources, equation (13) states that a properly definedcorrelation function gives the Green’s function plus itstime-reversed version, convolved with the autocorrela-tion of the source function.

In most practical situations, sources are not avail-able on a closed boundary. When a part of the inte-gration boundary is a free surface, the integrals needonly be evaluated over the remaining part of the bound-ary, hence, sources need only be available on an openboundary Ssrc that, together with the free surface,forms a closed boundary. This situation occurs in pas-

156 Wapenaar et al.

sive reflected-wave interferometry (section 6.2). In manyother situations the closed source boundary is replacedby an open surface, simply because the available sourceaperture is restricted to an open boundary. This is il-lustrated in Figure 2a for the situation of direct-waveinterferometry and in Figure 2b for controlled-sourcereflected-wave interferometry (these configurations willbe discussed in more detail in sections 6.1 and 5.1, re-spectively). Since the underlying representation (equa-tion 1) is of the correlation type (in which one of theGreen’s functions is backward propagating), radiationconditions do not apply anywhere on S, hence, replac-ing the closed surface by an open surface necessarilyleads to approximations. For the situation of direct-waveinterferometry (Figure 2a) the main effect is that thecorrelation function C(xB ,xA, t) is approximately pro-portional to G(xB ,xA, t) ∗ S(t), i.e, the time-reversedversion of the Green’s function in equation (13) is notrecovered (see e.g. Sabra et al. (2005b) and Miyazawaet al. (2008)). For reflected-wave interferometry (Fig-ure 2b), one-sided illumination can lead to severe dis-tortions of the retrieved Green’s function. Moreover,Snieder et al. (2006b) show that even for very simpleconfigurations it may lead to spurious multiples. An im-portant aspect of interferometry by MDD, discussed inthe next section, is that the approximations of one-sidedillumination are avoided in a natural way.

4 BASIC ASPECTS OFINTERFEROMETRY BYMULTI-DIMENSIONALDECONVOLUTION (MDD)

4.1 Simplification of the integral

The convolution-type Green’s function representation(equation 2) is a basic expression for seismic interfer-ometry by multi-dimensional deconvolution in open sys-tems. The right-hand side of this equation contains acombination of two convolution products. We show howwe can combine these two terms into a single term.

The Green’s function ˆG(xB ,x, ω) under the integral isthe unknown that we want to resolve by MDD. TheGreen’s function G(xB ,xS , ω) on the left-hand side andG(x,xS , ω) under the integral are related to the obser-vations. Hence, G is defined in the actual medium inside

as well as outside S, but for ˆG we are free to choose con-venient boundary conditions at S. In the following we let

S be an absorbing boundary for ˆG(xB ,x, ω), so that its

reciprocal ˆG(x,xB , ω) is outward propagating at S. Fur-thermore, we write G(x,xS , ω) as the superposition ofan inward- and outward-propagating part at x on S, ac-cording to G(x,xS , ω) = Gin(x,xS , ω) + Gout(x,xS , ω).Assuming S is sufficiently smooth, we replace the differ-ential operator ni∂i again by a pseudo-differential oper-ator ±jH1, this time for a medium with losses (Wape-

naar et al. 2001). The plus- and minus-sign in ±jH1

correspond to inward- and outward-propagating waves,respectively. Hence, we may write for the integral inequation (2), using the fact that H1 is symmetric,I

S

“(∂i

ˆGB)(GinS + Gout

S )− ˆGB∂i(GinS + Gout

S )”ni dx =I

S

“(−jH1

ˆGB)(GinS + Gout

S )− ˆGBjH1(GinS − Gout

S )”

dx

= 2

IS(−jH1

ˆGB)GinS dx = 2

IS(ni∂i

ˆGB)GinS dx,

(18)

where Gin/outS and ˆGB stand for Gin/out(x,xS , ω) and

ˆG(x,xB , ω) = ˆG(xB ,x, ω), respectively.

Using this in equation (2), assuming ρ is constanton S, we obtain

G(xB ,xS , ω) =

−2

jωρ

ISrec

`ni∂i

ˆG(xB ,x, ω)´Gin(x,xS , ω) dx. (19)

We added a subscript ‘rec’ in Srec to denote that theintegration surface contains the receivers of the Green’sfunction Gin. For convenience we introduce a dipoleGreen’s function, defined as

ˆGd(xB ,x, ω) =−2

jωρ

`ni∂i

ˆG(xB ,x, ω)´, (20)

so that equation (19) simplifies to

G(xB ,xS , ω) =

ISrec

ˆGd(xB ,x, ω)Gin(x,xS , ω) dx. (21)

In the underlying representation (equation 2) it was as-sumed that xB lies in V. In several applications of MDDxB is a receiver on Srec. For those applications we takexB just inside Srec to avoid several subtleties of takingxB on Srec. Moreover, in those applications it is oftenuseful to consider only the outward-propagating part ofthe field at xB . Applying decomposition at both sidesof equation (21) gives

Gout(xB ,xS , ω) =

ISrec

ˆGdout(xB ,x, ω)Gin(x,xS , ω) dx.

(22)Equation (22) is nearly the same (except for a differentnormalization) as our previously derived one-way repre-sentation for MDD (Wapenaar et al. 2008a). In the fol-lowing we continue with the notation of equation (21),

where G(xB ,xS , ω) and ˆGd(xB ,x, ω) may stand for thetotal or the outward-propagating fields at xB , depend-ing on the application. In most practical situations, re-ceivers are not available on a closed boundary, so theintegration in equation (21) is necessarily restricted toan open receiver boundary Srec. As long as the sourceposition xS is located at the appropriate side of Srec (i.e.,outside V), it suffices to take the integral over the openreceiver boundary: since the underlying representation


(equation 2) is of the convolution type, radiation condi-tions apply on the half-sphere that closes the boundary(assuming the half-sphere boundary is absorbing andits radius is sufficiently large), meaning that the con-tribution of the integral over that half-sphere vanishes.Hence, in the following we replace the closed bound-ary integral by an open boundary integral. In the timedomain equation (21) thus becomes

G(xB ,xS , t) =

ZSrec

Gd(xB ,x, t) ∗Gin(x,xS , t) dx. (23)

Unlike the correlation-type representation (equation 8),which holds under the condition that Ssrc is a closedsurface with very large radius, the convolution-type rep-resentation of equation (23) holds as long as the opensurface Srec is sufficiently smooth and xB and xS lie atopposite sides of Srec. Moreover, whereas equation (8)only holds for lossless media, equation (23) also holdsin media with losses. Equation (23) is an implicit repre-sentation of the convolution type for Gd(xB ,x, t). If itwere a single equation, the inverse problem would be ill-posed. However, equation (23) holds for each source po-sition xS (outside V), which we will denote from hereon

by x(i)S , where i denotes the source number. Solving the

ensemble of equations for Gd(xB ,x, t) involves MDD.

4.2 Transient sources

For practical applications the Green’s functions G andGin in equation (23) should be replaced by responsesof real sources, i.e., Green’s functions convolved withsource functions. For transient sources we may write forthe responses at x and xB

uin(x,x(i)S , t) = Gin(x,x

(i)S , t) ∗ s(i)(t), (24)

u(xB ,x(i)S , t) = G(xB ,x

(i)S , t) ∗ s(i)(t). (25)

Hence, by convolving both sides of equation (23) withs(i)(t) we obtain

u(xB ,x(i)S , t) =

ZSrec

Gd(xB ,x, t) ∗ uin(x,x(i)S , t) dx. (26)

Here u(xB ,x(i)S , t) and Gd(xB ,x, t) may stand for the

total or the outward-propagating fields at xB . We dis-cuss the modifications for noise sources in section 4.4and for simultaneous source acquisition in section 4.5.

Equation (26) is illustrated in Figure 3a for the situ-ation of direct-wave interferometry and in Figure 3b forreflected-wave interferometry (with superscripts “out”added). Note that we consider the sources to be irregu-larly distributed in space. This is possible because equa-tion (26) holds for each source separately whereas theintegration is performed along the receivers. The config-urations in Figures 3a and b will be discussed in moredetail in sections 6.1 and 5, respectively.

Although in the derivation of equation (26) we tac-itly assumed a point source of volume injection rate at

Bx

( )iSx

( )( , , )iB Su tx x

in ( )( , , )iSu tx x

d ( , , )BG tx x

recS

a)

V

x!!! !!!

in ( )( , , )iSu tx x

out ( )( , , )iB Su tx x

outd ( , , )BG tx x

Bx

( )iSx

recS

x!!! !!!

b)

V

recS

in ( )( , , )iSu tx x

in ( )( , , )iA Su tx x

( )iSx

c)

V

c)

( , , )A t! x x

Figure 3. (a) Illustration of the convolutional model (equa-

tion 26), underlying interferometry by MDD for the situa-tion of direct-wave interferometry. The inward-propagating

field uin(x,x(i)S , t) is convolved with Gd(xB ,x, t) and the

results for all receivers x are integrated along the receiver

boundary Srec, giving the response u(xB ,x(i)S , t). Assum-

ing equation (26) is available for many source positions

x(i)S , the Green’s function Gd(xB ,x, t) can be retrieved

by MDD. (b) Idem, for reflected-wave interferometry, with

u(xB ,x(i)S , t) and Gd(xB ,x, t) replaced by the outgoing fields

uout(xB ,x(i)S , t) and Gd

out(xB ,x, t), respectively. (c) Illustra-

tion of the point-spread function Γ(x,xA, t) (equation 29),defined as the crosscorrelation of the inward-propagating

fields at x and xA on Srec, summed over the sources. In theideal case, it is proportional to a band-limited delta functionon Srec, but in practice there are many factors that make it

different from a delta function.

158 Wapenaar et al.

x(i)S , equation (26) holds equally well for other types

of sources at x(i)S , such as volume force or disloca-

tion sources. The source also is not necessarily a pointsource. For an extended source, both sides of equation(26) can be integrated along the extended source, yield-ing an equation with exactly the same form, but withuin(x,x

(i)S , t) and u(xB ,x

(i)S , t) being the responses of

the extended source.

4.3 Relation with the correlation method

Solving equation (26) in a least-squares sense is equiva-lent to solving its normal equation (Menke 1989). Weobtain the normal equation by crosscorrelating bothsides of equation (26) with uin(xA,x

(i)S , t) (with xA on

Srec) and taking the sum over all sources (van der Neutet al. 2010). This gives

C(xB ,xA, t) =

ZSrec

Gd(xB ,x, t) ∗ Γ(x,xA, t) dx, (27)

where

C(xB ,xA, t) =Xi

u(xB ,x(i)S , t) ∗ uin(xA,x

(i)S ,−t)

=Xi

G(xB ,x(i)S , t) ∗Gin(xA,x

(i)S ,−t) ∗ S(i)(t), (28)

Γ(x,xA, t) =Xi

uin(x,x(i)S , t) ∗ uin(xA,x

(i)S ,−t)

=Xi

Gin(x,x(i)S , t) ∗Gin(xA,x

(i)S ,−t) ∗ S(i)(t), (29)

with

S(i)(t) = s(i)(t) ∗ s(i)(−t). (30)

Equation (27) shows that the correlation functionC(xB ,xA, t) is proportional to the sought Green’s func-tion Gd(xB ,x, t) with its source smeared in space andtime by Γ(x,xA, t).

Note that C(xB ,xA, t) as defined in equation (28)is a correlation function with a similar form as theone defined in equation (11). There are also some no-table differences. In equation (11) an integration takesplace along the source boundary Ssrc, whereas equation(28) involves a summation over individual sources. Ofcourse in practical situations the integral in equation(11) needs to be replaced by a summation as well, butthis should be done carefully, obeying the common re-strictions for discretization of continuous integrands. Onthe other hand, the summation in equation (28) simplytakes place over the available sources and as such putsno restrictions on the regularity of the source distribu-tion. We will hold on to the integral notation in equa-tion (11) versus the summation notation in equation(28) to express the differences in the assumptions on theregularity of the source distribution. The effect of thisdifference in assumptions will be illustrated in sections6.1 and 6.2. Another difference is that in equation (11)

the full wave fields at xA and xB are crosscorrelated,whereas in equation (28) the inward-propagating partof the wavefield at xA is crosscorrelated with the full (oroutward-propagating) wave field at xB . The final differ-ence is the filter F(xS , t) in equation (11) which shapesthe different autocorrelations of the different sources toan average autocorrelation function S(t); in equation(28) this filter is not needed.

Γ(x,xA, t) as defined in equation (29) is the cross-correlation of the inward-propagating wave fields at xand xA, summed over the sources (van der Neut et al.2010), see Figure 3c. We call Γ(x,xA, t) the interfer-ometric point-spread function (or plainly the point-spread function). If we would have a regular distribu-tion of sources with equal autocorrelation functions S(t)along a large planar source boundary, and if the mediumbetween the source and receiver boundaries were homo-geneous and lossless, the point-spread function wouldapproach a temporally and spatially band-limited deltafunction, see Appendix A for details. Hence, for this sit-uation equation (27) merely states that the correlationfunction C(xB ,xA, t) is proportional to the response ofa temporally and spatially band-limited source. In prac-tice, there are many factors that make the point-spreadfunction Γ(x,xA, t) deviate from a band-limited deltafunction. Among these factors are simultaneous sourceacquisition (section 4.5), medium inhomogeneities (sec-tion 5.1), multiple reflections in the illuminating wave-field (section 5.2), intrinsic losses (section 5.3), irregular-ity of the source distribution (sections 6.1 and 6.2), etc.For all those cases equation (27) shows that the point-spread function blurs the source of the Green’s functionin the spatial directions and generates ghosts (“spuriousmultiples”, Snieder et al. (2006b)) in the temporal direc-tion. MDD involves inverting equation (27), see section4.6. Ideally, this removes the distorting effects of thepoint-spread function Γ(x,xA, t) from the correlationfunction C(xB ,xA, t) and yields an improved estimateof the Green’s function Gd(xB ,x, t).

Note that the interferometric point-spread func-tion Γ(x,xA, t) plays a similar role in interferometry byMDD as the point-spread function (or spatial resolutionfunction) in optical, acoustical and seismic imaging sys-tems (Born & Wolf 1965; Norton 1992; Miller et al. 1987;Schuster & Hu 2000; Gelius et al. 2002; Lecomte 2008;Toxopeus et al. 2008; van Veldhuizen et al. 2008). Forexample, in seismic migration the point-spread functionis defined as the migration result of the response of asingle point scatterer. It is a useful tool to assess mi-gration results in relation with geological parameters,background model, acquisition parameters, etc. More-over, it is sometimes used in migration deconvolutionto improve the spatial resolution of the migration im-age (Hu et al. 2001; Yu et al. 2006). In a similar way,van der Neut & Thorbecke (2009) use the interfero-metric point-spread function Γ(x,xA, t) to assess thequality of the virtual source obtained by correlation-


based interferometry. Moreover, analogous to migrationdeconvolution, interferometry by MDD (i.e., inversionof equation (27)) aims at deblurring and deghosting thevirtual source. An important difference is that the inter-ferometric point-spread function is obtained from mea-sured responses, whereas the point-spread function usedin migration deconvolution is modeled in a backgroundmedium. As a result, the interferometric point-spreadfunction accounts much more accurately for the dis-torting effects of the medium inhomogeneities, includingmultiple scattering.

4.4 Noise sources

We show that equation (27) also holds for the situa-tion of simultaneously acting uncorrelated noise sources.To this end, we define the correlation function and thepoint-spread function, respectively, as

C(xB ,xA, t) = 〈u(xB , t) ∗ uin(xA,−t)〉, (31)

Γ(x,xA, t) = 〈uin(x, t) ∗ uin(xA,−t)〉, (32)

where the noise responses are defined as

uin(xA, t) =Xi

Gin(xA,x(i)S , t) ∗N (i)(t), (33)

uin(x, t) =Xj

Gin(x,x(j)S , t) ∗N (j)(t), (34)

u(xB , t) =Xj

G(xB ,x(j)S , t) ∗N (j)(t), (35)

in which the noise signals are assumed to be mutuallyuncorrelated, according to

〈N (j)(t) ∗N (i)(−t)〉 = δijS(i)(t). (36)

Upon substitution of equations (33) − (35) into equa-tions (31) and (32), using equation (36), it follows thatthe correlation function and the point-spread functionas defined in equations (31) and (32) are identical tothose defined in equations (28) and (29). Hence, whetherwe consider transient or noise sources, equation (27) isthe relation that needs to be inverted by MDD to resolvethe Green’s function Gd(xB ,x, t).

Note that the correlation function C(xB ,xA, t) asdefined in equation (31) resembles that defined in equa-tion (17). There are again three main differences. Equa-tion (17) is the correlation of noise responses which aredefined as integrals over sources on Ssrc (equations 14and 15), whereas the correlated responses in equation(31) are defined as summations over individual sources(equations 33 and 35). As mentioned before, this re-flects the difference in assumptions on the regularity ofthe source distribution (see sections 6.1 and 6.2). Fur-thermore, in equation (17) the full noise fields at xAand xB are crosscorrelated, whereas in equation (31) theinward-propagating part of the noise field at xA is cross-correlated with the full (or outward-propagating) noisefield at xB . Finally, an underlying assumption of equa-

! "iSx!! !!

in ! "! % % "mu tx !

Bx

! "m!

recS

x!!! !!!out ! "! % % "m

Bu tx !outd ! % % "BG tx x

V

Figure 4. Convolutional model underlying deblending by

MDD.

tion (17) is that the noise sources all have the same au-tocorrelation function S(t) (equation 16), whereas equa-tion (31) is still valid when the autocorrelations of thedifferent sources are different from one another (equa-tion 36).

4.5 Simultaneous source acquisition

A new trend in the seismic exploration community is si-multaneous source acquisition, also known as blendedacquisition (Beasley et al. 1998; Bagaini 2006; Ikelle2007; Stefani et al. 2007; Howe et al. 2008; Hampsonet al. 2008; Berkhout 2008). Seismic sources are firedwith relatively small intervals one after the other, to re-duce the total acquisition time. As a result, the seismicresponse can be seen as a superposition of time-delayedseismic shot records. Using standard shot-record ori-ented processing and imaging, “crosstalk” between thesources causes the images to be noisy. The crosstalkcan be reduced by using phase-encoded sources (Bagaini2006; Ikelle 2007) or simultaneous noise sources (Howeet al. 2008), by randomizing the time interval betweenthe shots (Stefani et al. 2007; Hampson et al. 2008), orby inverting the “blending operator” (Berkhout 2008).Here we briefly discuss deblending as a form of seismicinterferometry by MDD.

Consider the configuration depicted in Figure 4,where σ(m) denotes a group of source positions x

(i)S .

Although the figure suggests that these sources are ad-jacent to each other, they may also be randomly selectedfrom the total array of sources. Assuming the sourceat x

(i)S emits a delayed source wavelet s(i)(t − ti), the

blended inward- and outward-propagating fields at Srec

160 Wapenaar et al.

are given by

uin(x, σ(m), t)=X

x(i)S∈σ(m)

Gin(x,x(i)S , t) ∗ s(i)(t− ti),

(37)

uout(xB , σ(m), t)=X

x(i)S∈σ(m)

Gout(xB ,x(i)S , t)∗s(i)(t−ti),

(38)

where x(i)S ∈ σ(m) denotes that the summation takes

place over all source positions x(i)S in group σ(m). By

convolving both sides of equation (23) with s(i)(t − ti)and summing over all sources in σ(m) we obtain, anal-ogous to equation (26),

uout(xB , σ(m), t) =

ZSrec

Goutd (xB ,x, t)∗uin(x, σ(m), t) dx.

(39)When the receivers on Srec are real receivers in a bore-hole, then uout(xB , σ(m), t) and uin(x, σ(m), t) are (de-composed) measured blended wave fields in the bore-hole. On the other hand, in case of surface data ac-quisition, uout(xB , σ(m), t) represents the blended dataafter model-based receiver redatuming to Srec anduin(x, σ(m), t) represents the blended sources, forwardextrapolated through the model to Srec.

Analogous to section 4.3 we obtain the normalequation by crosscorrelating both sides of equation (39)with uin(xA, σ(m), t) (with xA on Srec) and taking thesum over all source groups σ(m). This gives

C(xB ,xA, t) =

ZSrec

Goutd (xB ,x, t) ∗ Γ(x,xA, t) dx,

(40)where

C(xB ,xA, t) =Xm

uout(xB , σ(m), t) ∗uin(xA, σ(m),−t),

(41)

Γ(x,xA, t) =Xm

uin(x, σ(m), t) ∗ uin(xA, σ(m),−t).

(42)The correlation function defined in equation (41) is sim-ilar to that in equation (28), except that it is contam-inated by crosstalk between the sources within eachsource group σ(m). The point-spread function definedin equation (42) also contains crosstalk contributions.In section 5.1 we present some examples of this point-spread function. Deblending involves inverting equation(40) by MDD. Ideally this eliminates the crosstalk fromthe correlation function and gives the deblended virtualsource response Gout

d (xB ,x, t).

The discussed representations for blended datahave two interesting limiting cases. When each sourcegroup σ(m) consists of a single source, then we obtainthe expressions for transient sources discussed in sec-tions 4.2 and 4.3. On the other hand, when there isonly one source group containing all sources and when

the source wavelets are replaced by mutually uncorre-lated noise signals, then we obtain the expressions fornoise sources discussed in section 4.4.

4.6 Resolving the Green’s function

Interferometry by MDD essentially involves inversion ofequation (27) (or 40). In general, the existence of theinverse of the point-spread function is not guaranteed.Recall from equations (29), (32) and (42) that the point-spread function (explicitly or implicitly) involves a sum-mation over source positions. To evaluate this function,the source positions do not need to be known and thesource distribution does not need to be regular, butclearly the well-posedness of its inverse depends on thenumber of available sources, the total source apertureand the source bandwidth. In practical situations a spec-tral analysis of the point-spread function helps to assessfor which spatial and temporal frequencies the inversioncan be carried out (van der Neut et al. 2011).

For the actual inversion of equation (27) (or 40) wetransform this equation to the frequency domain andreplace the integration along the receivers by a summa-tion, according to

C(xB ,x(l)A , ω) =

Xk

ˆGd(xB ,x(k), ω)Γ(x(k),x(l)A , ω),

(43)

for all x(l)A on the receiver surface Srec. Note that this

discretization assumes a regular sampling of the receivercoordinate x(k). This is a less severe restriction thanthat for discretizing a source integral (like the one inequation (11)), because the receivers are often well sam-pled and their positions are usually known. In case ofirregular receiver sampling, a regularization procedure(Duijndam et al. 1999) could be applied prior to invert-ing equation (43). The system of equations (43) can be

solved for ˆGd(xB ,x(k), ω) for each frequency componentseparately. In practice this is done by a stabilized ma-trix inversion per frequency component, while takingcare of the limitations discussed above. Transformingthe end-result back to the time domain gives a band-limited estimate of Gd(xB ,x(k), t), which completes theMDD process.

It is beyond the scope of this paper to discuss thenumerical aspects of the matrix inversion. The inver-sion is similar to that proposed by Berkhout (1982)and Wapenaar & Berkhout (1989) in the context ofmodel-based inverse seismic wavefield extrapolation andimaging. Tanter et al. (2000) apply matrix inversionto a measured wave-field propagator in the context ofthe time-reversal method and thus improve the sourceimaging capabilities of that method. Whereas Berkhout(1982) proposes least-squares inversion, Tanter et al.(2000) employ singular-value decomposition to invertthe matrix. Similarly, in seismic interferometry by MDDthe matrix inversion can be done by least-squares inver-


sion (Wapenaar et al. 2008b) or by singular-value de-composition (Minato et al. 2011). We refer to van derNeut et al. (2011) for a discussion on the stability as-pects of interferometry by MDD, and to Ruigrok et al.(2010) and Hunziker et al. (2011) for a discussion on thesampling and illumination aspects.

4.7 Vectorial fields

Until now we considered scalar fields. For vectorial fieldsthe representations and algorithms are straightforwardextensions of those discussed above. Here we discussthe main modifications. The convolutional representa-tion (26) is for vectorial fields extended to

u(xB ,x(i)S , t) =

ZSrec

Gd(xB ,x, t) ∗ uin(x,x(i)S , t) dx (44)

(the integrand is a convolutional matrix-vector prod-

uct). Here uin(x,x(i)S , t) and u(xB ,x

(i)S , t) are the in-

ward propagating and total (or outward-propagating)field vectors, respectively, due to transient sources atx

(i)S (in the case of simultaneous source acquisition,

the source positions x(i)S need to be replaced by source

groups σ(m)). For example, for the situation of elasto-dynamic waves the field vectors are defined as

uin =

0@Φin

Ψin

Υin

1A and u =

0@ΦΨΥ

1A , (45)

where Φ, Ψ and Υ represent P , S1 and S2 waves, respec-tively. In practice these fields are obtained by applyingdecomposition of multi-component data. Moreover, forthis situation, Gd(xB ,x, t) can be written as

Gd(xB ,x, t) =

0@Gφ,φd Gφ,ψ

d Gφ,υd

Gψ,φd Gψ,ψ

d Gψ,υd

Gυ,φd Gυ,ψ

d Gυ,υd

1A (xB ,x, t),

(46)where the Green’s function Gp,q

d (xB ,x, t) is the dipoleresponse (analogous to equation (20)) of the mediumin V (with absorbing boundary conditions at Srec) interms of an inward-propagating q-type wave field atsource position x and a total (or outward-propagating)p-type wave field at receiver position xB (Wapenaar &

Berkhout 1989). When the sources at x(i)S are multi-

component sources, equation (44) can be extended to

U(xB ,x(i)S , t) =

ZSrec

Gd(xB ,x, t) ∗Uin(x,x(i)S , t) dx,

(47)where Uin and U are matrices of which the columns con-tain field vectors uin and u for different source compo-nents (similar as the different columns of Gd are relatedto different source types). Equations (44) and (47) arenot restricted to the elastodynamic situation but applyto any vectorial wave or diffusion field, including electro-

magnetic, poroelastic and seismoelectric fields (Wape-naar et al. 2008a).

Interferometry by MDD means that equation (47)needs to be solved for Gd(xB ,x, t). Similar as for thescalar situation, the well-posedness of this inverse prob-lem depends on the number of available sources, thesource aperture, the bandwidth, and, in addition, thenumber of independent source components. Since theGreen’s matrix is resolved by MDD, the sources do notneed to be regularly distributed and the source compo-nents do not need to be mutually orthonormal.

Similar as in the scalar case, to obtain the nor-mal equation for least-squares inversion, we crosscor-relate both sides of equation (47) with the transposed

of Uin(xA,x(i)S , t) (with xA on Srec) and take the sum

over all sources, according to

C(xB ,xA, t) =

ZSrec

Gd(xB ,x, t)∗Γ(x,xA, t) dx, (48)

where

C(xB ,xA, t) =Xi

U(xB ,x(i)S , t) ∗ {Uin(xA,x

(i)S ,−t)}t,

(49)

Γ(x,xA, t) =Xi

Uin(x,x(i)S , t) ∗ {Uin(xA,x

(i)S ,−t)}t,

(50)with superscript t denoting transposition. HereC(xB ,xA, t) and Γ(x,xA, t) are the correlation andpoint-spread functions, respectively, in matrix form.Equation (48) shows again that the correlation func-tion is proportional to the sought Green’s functionGd(xB ,x, t) with its source smeared in space and timeby the point-spread function.

For the situation of noise sources the correlationand point-spread matrices are given by

C(xB ,xA, t) = 〈u(xB , t) ∗ {uin(xA,−t)}t〉, (51)

Γ(x,xA, t) = 〈uin(x, t) ∗ {uin(xA,−t)}t〉, (52)

where, analogous to equations (33) − (35), the field vec-tors uin and u are the responses of simultaneously actingmutually uncorrelated multi-component noise sources.It can be shown in a similar way as in section 4.4 thatthe correlation and point-spread functions defined inequations (51) and (52) are identical to those definedin (49) and (50). Hence, whether we consider transientor noise sources, equation (48) is the relation that needsto be inverted by MDD to resolve the Green’s functionGd(xB ,x, t). This is done in a similar way as describedin section 4.6 for the scalar situation.

162 Wapenaar et al.

x1 (m)

x 3 (m)

!2000 !1000 0 1000 2000

200

400

600

800

1000

c (m/s)

1000

1500

2000

2500

3000a)

b)!480 !240 0 240 4800

0.2

0.4

0.6

0.8

1

Horizontal offset (m)

Tim

e (s

)

!480 !240 0 240 4800

0.2

0.4

0.6

0.8

1


Tim

e (s

)

c)

!480 !240 0 240 480!1

!0.5

0

0.5

1


Tim

e (s

)

d)

!480 !240 0 240 4800

0.2

0.4

0.6

0.8

1


Tim

e (s

)

e)

Figure 5. (a) Configuration for the “virtual source method”.

The sources are situated at the surface (black dots) andthe receivers in a horizontal borehole (the green triangles

at x3 = 500m) below a complex overburden. (b) Modeled

response of a source at the center of the surface. The am-plitudes below the dashed line have been multiplied by a

factor 3. (c) Result of interferometry by crosscorrelation of

decomposed wave fields (red), compared with directly mod-eled reflection response of a source at the center of the bore-

hole (black). (d) Point-spread function (clipped at 20% of its

maximum amplitude). (e) MDD result (red) compared withmodeled response (black).

5 APPLICATIONS INCONTROLLED-SOURCEINTERFEROMETRY

Unlike ambient-noise interferometry, controlled-sourceinterferometry is usually applied to situations in whichthe sources are regularly distributed along a surface(Schuster 2009). Hence, for controlled-source interfer-ometry, MDD is usually not required to correct for anirregular source distribution. However, there are sev-eral other limiting factors in controlled-source inter-ferometry for which MDD may provide a solution. Inthe following sections we show with numerical exampleshow MDD accounts for the effects of medium inhomo-geneities in the “virtual source method” (section 5.1),for multiples in the illuminating wave field in ocean-bottom multiple elimination (section 5.2) and for lossesin CSEM interferometry (section 5.3).

5.1 “Virtual source method”

Although creating a virtual source is the essence of allseismic interferometry methods, in the seismic explo-ration literature the term “virtual source method” is of-ten synonymous with the method developed by Bakulin& Calvert (2004, 2006). Figure 2b shows the basic con-figuration. Sources are present at the Earth’s surface,denoted by Ssrc. The responses of these sources are mea-sured in a near-horizontal borehole, below a complexoverburden. The responses u(xA,xS , t) and u(xB ,xS , t)at any combination of two receivers xA and xB in theborehole are crosscorrelated and the results are inte-grated along the sources xS at Ssrc, according to equa-tion (11). The correlation function C(xB ,xA, t) is inter-preted as the response at xB of a virtual source at xA.Because this virtual source is situated below the com-plex overburden, its response contains less-distorted re-flections of the deeper target than the original responsesof the sources at the surface Ssrc.

From the theory it follows that this method involvesapproximations. If Ssrc were a closed surface, the corre-lation function C(xB ,xA, t) would be proportional to{G(xB ,xA, t) + G(xB ,xA,−t)} ∗ S(t) (equation 13), inwhich G(xB ,xA, t)∗S(t) is indeed the response of a vir-tual source at xA. However, because Ssrc is limited to apart of the Earth’s surface, the correlation function isa distorted version of G(xB ,xA, t) ∗ S(t) and containsspurious multiples (Snieder et al. 2006b). To compen-sate for this, Bakulin & Calvert (2004) apply a timewindow around the first arrival of u(xA,xS , t). Bakulin& Calvert (2006) propose to replace the correlation by atrace-by-trace deconvolution for the downgoing part ofu(xA,xS , t). They show that this approach compensatesfor variations in the source wavelet and, partly, for re-verberations in the overburden. The spurious multiplescan be further suppressed by applying up/down decom-position to u(xB ,xS , t) as well as u(xA,xS , t) (Mehtaet al. 2007a; van der Neut & Wapenaar 2009).


!240 !120 0 120 240!1

!0.5

0

0.5

1


Tim

e (s

)

a)

!240 !120 0 120 240!1

!0.5

0

0.5

1


Tim

e (s

)

b)

Figure 6. (a) Point-spread function for blended data from64 groups of four sources each, with a regular time interval

of 0.25 s. (b) Idem, after adding random variations to the

time interval.

We illustrate the correlation approach with a nu-merical example. Figure 5a shows the configuration,with a regular distribution of sources at the surface (de-noted by the black dots) and receivers in a horizontalborehole (the green triangles at x3 = 500 m). The up-per part of the overburden, i.e., the medium between thesources and the receivers, contains significant variationsof the medium parameters in the lateral as well as in thevertical direction. Below the receivers there is a reservoirlayer (the orange layer in Figure 5a). The lower part ofthe overburden, i.e., the medium between the receiversin the borehole and the reservoir, is homogeneous (thelight-green layer). Figure 5b shows a modeled responseof a source at the center of the surface, observed by thereceivers in the borehole. The modeling was done with-out free-surface effects. The response of the reservoiris denoted by the arrow. It is distorted by the inho-mogeneities of the overburden. The aim of the “virtualsource method” is to reduce these distortions. We followthe approach of Mehta et al. (2007a), hence, we first ap-ply decomposition of the field observed in the boreholeinto downgoing and upgoing waves. This requires thatmulti-component data are available (in this case theacoustic pressure and the normal particle velocity com-ponent). We call the decomposed fields uin(xA,x

(i)S , t)

and uout(xB ,x(i)S , t), respectively. The correlation func-

tion is defined, analogous to equation (28), as

C(xB ,xA, t) =Xi

uout(xB ,x(i)S , t) ∗ uin(xA,x

(i)S ,−t).

(53)The virtual source at xA is chosen at the center of

the borehole whereas the receiver coordinate xB variesalong the borehole. Figure 5c shows this correlationfunction in red. The directly modeled reflection responseof a source for downgoing waves at xA is shown in blackin this same figure. Note that the retrieved reservoir re-sponse matches the modeled response reasonably well,but still we observe some distortions along this reflectionevent as well as some spurious events at later times.

Interferometry by MDD provides a more accurateway to retrieve the virtual source response. Figure 3bshows the configuration. Assuming a regular source dis-tribution, this configuration corresponds to that of Fig-ure 2b, except that Figure 3b illustrates the convolu-tional representation that underlies MDD. The responseuout(xB ,x

(i)S , t) is defined, analogous to equation (26),

as

uout(xB ,x(i)S , t) =

ZSrec

Gdout(xB ,x, t)∗uin(x,x

(i)S , t) dx,

(54)i.e., the convolution of the inward propagating wave fielduin(x,x

(i)S , t) with the Green’s function Gd

out(xB ,x, t),integrated along the receivers x in the borehole. Thisborehole is denoted as Srec. Our aim is to retrieveGd

out(xB ,x, t) by inverting, analogous to equation (27),the normal equation

C(xB ,xA, t) =

ZSrec

Gdout(xB ,x, t) ∗ Γ(x,xA, t) dx.

(55)The correlation function C(xB ,xA, t) at the left-handside was already shown in Figure 5c. The point-spread function Γ(x,xA, t) at the right-hand side isobtained, according to equation (29), by crosscorrelat-

ing uin(x,x(i)S , t) with uin(xA,x

(i)S , t) and summing over

the sources. It is shown in Figure 5d for fixed xA andvariable x. Note that it deviates significantly from theband-limited delta function in Figure A1, due to theinhomogeneities of the overburden. According to equa-tion (55) this point-spread function smears the sourceof Gd

out(xB ,x, t) in space and time, which explains thedistortions along the reflection event in Figure 5c. Byapplying MDD, that is, deconvolving the correlationfunction in Figure 5c for the point-spread function ofFigure 5d, we obtain an estimate of the Green’s func-tion Gd

out(xB ,x, t). The result (for x = xA fixed at thecentral geophone and variable xB) is shown in Figure 5eby the red traces (for display purposes convolved withS(t)), where it is compared with the directly modeledresponse (black traces). Note that the match is nearlyperfect and that the spurious events vanished, whichconfirms that in this example MDD properly correctsfor the inhomogeneities of the overburden.

The elastodynamic extension of this approachyields improved virtual P - and S-wave source responseswith reliable amplitudes along the reflection events(van der Neut et al. 2011), suited for quantitative

164 Wapenaar et al.

Bx

( )iSx

recS

x!!! !!!

V

outd ( , , )BG tx x

out ( )( , , )iB Su tx x

in ( )( , , )iSu tx x

Figure 7. Convolutional model for ocean-bottom data with

multiples.

amplitude-versus-angle (AVA) inversion. A further dis-cussion is beyond the scope of this paper.

We conclude this section by presenting the point-spread function for the situation of simultaneous sourceacquisition. We consider a similar configuration as inFigure 5a, except that for simplicity the propagationvelocity is taken constant at 2000 m/s. There are 256sources at the surface with a lateral spacing of 20m. We form 64 source groups σ(m), each containingfour adjacent sources which emit transient wavelets,0.25 s after one another. The inward-propagating fielduin(x, σ(m), t) at the receiver array at x3 = 500 mis defined by equation (37). The point-spread functionΓ(x,xA, t), as defined in equation (42), is shown in Fig-ure 6a, with xA fixed at the center of the array andx variable along the array. Note that the band-limiteddelta function around x = xA and t = 0 in Figure 6aresembles that in Figure A1, because the medium ishomogeneous in both cases. However, Figure 6a con-tains in addition a number of temporally and spatiallyshifted band-limited delta functions, as a result of thecrosstalk between the sources within each source groupσ(m). These shifted delta functions explain the crosstalkin the correlation function C(xB ,xA, t) via equation(40). To obtain the deblended virtual source responseGd

out(xB ,x, t) from C(xB ,xA, t), equation (40) needsto be inverted (not shown).

As mentioned in section 4.5, the crosstalk asso-ciated to standard shot-record oriented processing ofblended data is sometimes reduced by randomizing thetime interval between the shots. We add random varia-tions (uniform between + and −50%) to the 0.25 s timeinterval and evaluate again the point-spread functionΓ(x,xA, t), see Figure 6b. Note that the band-limiteddelta function around x = xA and t = 0 remains in-tact, whereas the crosstalk disperses in space and time.Inverting noisy point-spread functions like the one inFigure 6b may be a more stable process than invertingnearly periodic point-spread functions like the one inFigure 6a. A further discussion of deblending is beyondthe scope of this paper.

5.2 Ocean-bottom multiple elimination

As already mentioned in the introduction, interferome-try by MDD is akin to multiple elimination of ocean-bottom data (Amundsen 2001). Here we briefly re-view ocean-bottom multiple elimination, explained asan interferometry-by-MDD process. Figure 7 shows theconvolutional model of Figure 3b, modified for the situ-ation of ocean-bottom data, including multiple reflec-tions. Equation (54) gives the relation between theinward- and outward-propagating fields at the oceanbottom and the Green’s function Gd

out(xB ,x, t). Re-call that the bar denotes a reference situation withpossibly different boundary conditions at Srec and/ordifferent medium parameters outside V. For the refer-ence situation we choose an absorbing ocean bottomSrec and a homogeneous upper half-space, which impliesthat Gd

out(xB ,x, t) is the response of the half-space be-low Srec, without any multiple reflections related to theocean bottom and the water surface. Interferometry byMDD resolves Gd

out(xB ,x, t) by inverting equation (55).We illustrate this with a numerical example. Fig-

ure 8a shows the configuration, with sources just be-low the water surface and multi-component receivers(acoustic pressure and normal particle velocity) at theocean bottom. The lower half-space contains a reservoirlayer. Figure 8b shows the modeled response of the cen-tral source, observed at the ocean bottom. The directwave and the reservoir response (the latter denoted byan arrow) are clearly distinguishable, as well as manymultiple reflections. Decomposition is applied to themulti-component data at the ocean bottom, using themedium parameters of the first layer below the oceanbottom (Amundsen & Reitan 1995; Schalkwijk et al.2003). This gives the inward and outward-propagating

waves uin(xA,x(i)S , t) and uout(xB ,x

(i)S , t) just below the

ocean bottom. The correlation function, defined in equa-tion (53), is shown in Figure 8c (red traces), and is com-pared with the directly modeled response of the reser-voir layer (black traces). Note that the match of thecorrelation function with the reservoir response is quitegood, but the correlation function contains in additionmany spurious multiple reflections that are not presentin the reservoir response. The amplitudes of these spu-rious multiple reflections, which appear below the hori-zontal dashed line in Figure 8c, have been multipliedby a factor 3 for display purposes (in more compli-cated situations these spurious multiples would interferewith the retrieved primaries). The point-spread func-tion Γ(x,xA, t), defined in equation (29), is shown inFigure 8d. Apart from the band-limited delta function,it contains multiple reflections. According to equation(55), the correlation function in Figure 8c can be seen asthe reservoir response Gd

out(xB ,x, t) convolved in spaceand time with the point-spread function in Figure 8d.This explains the spurious multiples in Figure 8c. Thesespurious multiples are removed by deconvolving for thepoint-spread function, i.e., by inverting equation (55)


( )iSx

recSV

3Water: 1500 m/s, 1000 kg/mc !" "

0 m

500 m

760 m

880 m

a)

3Overburden: 2000 m/s, 1300 kg/mc !" "

3Reservoir: 2500 m/s, 1800 kg/mc !" "

3Half-space: 2200 m/s, 1200 kg/mc !" "

b)!640 !320 0 320 6400

0.5

1

1.5

Offset (m)

Tim

e (s

)

!640 !320 0 320 6400

0.5

1

1.5


Tim

e (s

)

c)

!640 !320 0 320 640

!1

!0.5

0

0.5

1


Tim

e (s

)

d)

!640 !320 0 320 6400

0.5

1

1.5


Tim

e (s

)

e)

Figure 8. (a) Configuration for ocean-bottom multiple elim-

ination. The sources are situated just below the free surfaceand the receivers at the ocean bottom. (b) Modeled response

of a source at the center of the surface. (c) Result of in-terferometry by crosscorrelation of decomposed wave fields

(red), compared with directly modeled reflection response

of a source at the center of the ocean bottom (black). Theamplitudes below the dashed line have been multiplied bya factor 3. (d) Point-spread function (clipped at 20% of its

maximum amplitude). (e) MDD result (red) compared withmodeled response (black). The amplitudes below the dashed

line have been multiplied by a factor 3

for Gdout(xB ,x, t). The result of this MDD procedure

(for x = xA fixed at the central geophone and variablexB) is shown in Figure 8e by the red traces (for displaypurposes convolved with S(t)), and is compared withthe directly modeled response (black traces). Note thatthe spurious multiples of Figure 8c have been very wellsuppressed (for display purposes the amplitudes of themultiple residuals below the dashed line in Figure 8ehave been multiplied with the same factor as the spuri-ous multiples in Figure 8c).

We included this example to show the relation be-tween interferometry by MDD and the methodology ofocean-bottom multiple elimination. In the example insection 5.1 we showed that interferometry by MDD ac-counts for overburden distortions, but in that examplethe effect of multiple reflections was small. When bore-hole data are distorted by overburden effects as well asmultiple reflections, interferometry by MDD simultane-ously accounts for both types of distortions.

5.3 CSEM interferometry

In controlled-source electromagnetic (CSEM) prospect-ing, a low-frequency source above the ocean bottomemits a diffusive EM field into the subsurface, of whichthe response is measured by multi-component EM re-ceivers at the ocean bottom (Figure 9a). Although thespatial resolution of CSEM data is much lower thanthat of seismic data, an important advantage of CSEMprospecting is its potential to detect a hydrocarbon ac-cumulation in a reservoir due to its high conductivitycontrast (Ellingsrud et al. 2002; Moser et al. 2006). Fig-ure 9b shows a modeled 2D CSEM response of an inlineelectric current source of 0.5 Hz. The source is posi-tioned at the center of the array, 50 m above the oceanbottom, see Figure 9a. This response represents the in-line electric field component at the ocean bottom, as afunction of source-receiver offset. The receiver samplingis 20 m and the total length of the array is 10 km.

Because a CSEM measurement is nearly monochro-matic, the response of a reservoir cannot be separated intime from other events. As a matter of fact, it is largelyovershadowed by the direct field and the response of theairwave. Amundsen et al. (2006) show that decomposi-tion of CSEM data into downward and upward decayingfields improves the detectability of hydrocarbon reser-voirs. Here we show that the combination of decomposi-tion and interferometry by MDD yields accurate quan-titative information about the reservoir response (Slobet al. 2007; van den Berg et al. 2008).

Decomposition requires that multi-component dataare available. For the example of Figure 9, this meansthat apart from the inline electric field data (Figure 9b),transverse magnetic data are required as well. Usinga decomposition algorithm similar as that for seismicdata, CSEM data can be decomposed into downwardand upward decaying fields (Ursin 1983; Amundsen

166 Wapenaar et al.

( )iSx

recSV

12rAir: 10 S/m, 1! "#$ $

rWater: 3 S/m, 80! "$ $

rReservoir: 0.02 S/m, 2.1! "$ $

rOverburden: 0.5 S/m, 17! "$ $

rHalf-space: 0.5 S/m, 17! "$ $

950 m

0 m

1000 m

1200 m

1250 m

a)

!6000 !4000 !2000 0 2000 4000 600010!10

10!8

10!6

10!4

Ampl

itude

(Vs/

m)

Offset (m)

b)

!6000 !4000 !2000 0 2000 4000 60000

0.2

0.4

0.6

0.8

1

Norm

alize

d Am

plitu

de

Offset (m)

c)

!6000 !4000 !2000 0 2000 4000 6000!0.5

0

0.5

1

1.5

2

2.5

3x 10!6

Ampl

itude

Offset (m)

d)

!6000 !4000 !2000 0 2000 4000 60000

0.5

1

1.5

2

2.5

3

3.5x 10!4

Ampl

itude

Offset (m)

e)

Figure 9. (a) Configuration for CSEM interferometry. Thesources just above the ocean bottom are inline electric cur-

rent sources, emitting monochromatic diffusive EM fields intothe subsurface. The receivers at the ocean bottom measure

the inline electric field and the transverse magnetic field. (b)

Inline electric field at ocean bottom. (c) Result of interferom-etry by crosscorrelation (red), compared with directly mod-

eled CSEM reservoir response (black). (d) Point-spread func-

tion. (e) MDD result (red) compared with modeled response(black).

et al. 2006). We designate these fields as uin(xA,x(i)S , ω)

and uout(xB ,x(i)S , ω), respectively, where superscripts

“in” and “out” denote that at the receiver surface Srec

in Figure 9a these fields diffuse inward to and outwardfrom V, respectively. Note that ω is considered constant(ω = 2πf , with f = 0.5 Hz).

Because in CSEM we consider monochromaticfields, we replace equations (55), (53) and (29) by thefollowing frequency-domain expressions

C(xB ,xA, ω) =

ZSrec

ˆGdout(xB ,x, ω)Γ(x,xA, ω) dx,

(56)where the correlation function and the point-spreadfunction are defined as

C(xB ,xA, ω) =Xi

uout(xB ,x(i)S , ω){uin(xA,x

(i)S , ω)}∗,

(57)

Γ(x,xA, ω) =Xi

uin(x,x(i)S , ω){uin(xA,x

(i)S , ω)}∗,

(58)

respectively. ˆGdout(xB ,x, ω) in equation (56) is the

sought reservoir response that would be obtained witha monochromatic source for an inward diffusing field atx and a receiver for an outward diffusing field at xB(both at Srec), in a configuration with a reflection-freeocean bottom. The monochromatic correlation func-tion defined in equation (57) is obtained by correlat-ing the inward and outward diffusing fields at Srec.The result is shown in Figure 9c (red curve), whereit is compared with the directly modeled responseˆGd

out(xB ,x, ω) (black curve), with x = xA at the centerof the array at Srec. The differences are mainly due tothe dissipation of the conducting water layer, for whichno compensation takes place in the correlation method.For ease of comparison both responses were normalizedso that the maxima of both curves are the same. Thedeviations at intermediate offsets hinder the quantifica-tion of the reservoir parameters.

Figure 9d shows the monochromatic point-spreadfunction, defined in equation (58). It deviates from aband-limited spatial delta function due to the dissipa-tion of the water layer as well as the interactions withthe water surface. MDD involves inversion of equation(56), i.e., removing the effect of the point-spread func-tion of Figure 9d from the correlation function in Figure9c. The result is shown in Figure 9e (red curve), whereit perfectly matches the directly modeled result (blackcurve). Note that, unlike in Figure 9c, no normalizationwas needed here.

This example shows that, at least in principle, in-terferometry by MDD compensates for dissipation andthus leads to an accurate retrieval of the CSEM reser-voir response. The effects of the conducting water layerare completely eliminated, including those of the directfield and the airwave. The final response is independentof the water depth and contains quantitative informa-


120 oW 100 oW 80 oW 60 oW

20 oN

30 oN

40 oN

50 oN

Longitude [deg]

Latit

ude

[deg

]

a)

10!3 10!2 10!1 1002800

3000

3200

3400

3600

3800

4000

4200

4400

Frequency (Hz)

VR

(m/s

)

b)

0 500 1000 1500 2000

0

500

1000

1500

Distance along South-North array (km)

Tim

e (s

)

c)

Figure 10. Model for passive surface-wave interferometry.(a) Map of USA, with USArray stations (green triangles),storm-related microseismic sources (blue dots), and a scat-

terer (black dot). (b) Rayleigh-wave dispersion curve (fun-damental mode), based on the PREM model. (c) Modeled

Rayleigh-wave response along South-North array in central

USA.

tion about the reservoir layer. It should be noted thatwe considered an ideal situation of well-sampled data,measured with high precision and no noise added. Fora more detailed discussion of CSEM interferometry byMDD, see Hunziker et al. (2009, 2011) and Fan et al.(2009).

6 APPLICATIONS IN AMBIENT-NOISEINTERFEROMETRY

In ambient-noise interferometry, the sources are un-known and usually irregularly distributed. MDD has thepotential to compensate for the source irregularity. Weillustrate this aspect with numerical examples for pas-sive surface-wave interferometry (section 6.1) and forpassive body-wave interferometry (section 6.2).

6.1 Passive surface-wave interferometry

One of the most widely used applications of direct-waveinterferometry is the retrieval of seismic surface wavesbetween seismometers from ambient noise (Campillo &Paul 2003; Shapiro & Campillo 2004; Shapiro et al.2005; Sabra et al. 2005a,b; Larose et al. 2006; Ger-stoft et al. 2006; Yao et al. 2006; Bensen et al. 2007,2008; Yao et al. 2008; Gouedard et al. 2008; Liang &Langston 2008; Ma et al. 2008; Lin et al. 2009; Pi-cozzi et al. 2009). Usually one considers the retrievalof the fundamental mode only. It is well known thatin a layered medium, surface waves consist of a funda-mental mode and higher-order modes (e.g. Nolet (1975);Gabriels et al. (1987)). Halliday & Curtis (2008) andKimman & Trampert (2010) carefully analyze interfer-ometry of surface waves with higher-order modes. Theyshow that, when the primary sources are confined to thesurface, crosscorrelation gives rise to spurious interfer-ences between higher-order modes and the fundamentalmode, whereas the presence of sources at depth, as pre-scribed by the theory (Wapenaar & Fokkema 2006), en-ables the correct recovery of all modes independently. Inrealistic situations the source distribution is mainly con-fined to the surface, which explains why in the currentpractice of surface-wave interferometry only the funda-mental modes are properly retrieved.

In agreement with the current practice of surface-wave interferometry, in the following numerical exam-ple we consider fundamental Rayleigh-wave modes only.These modes can be treated as the solution of a scalar2D wave equation with the propagation velocity be-ing the dispersive Rayleigh wave velocity of the layeredmedium. Hence, for interferometry we can make use ofthe 2D version of the scalar Green’s function represen-tations, discussed in the theory sections. The basic con-figuration is shown in Figure 2a for the crosscorrelationmethod, and in Figure 3a for the MDD method. Forthe current purpose these figures should be seen as planviews.

Consider Figure 10a, which shows a map of theUSA, 38 receiver stations of the USArray (green tri-angles) and 242 sources along the East coast (bluedots), for example representing storm-related microseis-mic sources (Bromirski 2001). Moreover, the black dotdenotes a strong scatterer, representing the Yellowstoneplume (Smith et al. 2009). Assuming a layered medium,we compute the dispersion curve of the fundamentalmode of the Rayleigh-wave for the upper 300 km ofthe PREM model (Dziewonski & Anderson 1981), us-ing the approach described by Wathelet et al. (2004).The dispersion curve is shown in Figure 10b. We definethe sources as simultaneously acting uncorrelated noisesources, with a central frequency of 0.04 Hz. Using thecomputed dispersion curve, we model the surface-waveresponse of the distribution of noise sources at all indi-cated receivers. Figure 10c shows 1500 s of the responsealong the indicated South-North array in central USA

168 Wapenaar et al.

400600800100012001400160018002000

0

100

200

300

400

500

600

700

800

Distance to virtual source (km)

Tim

e (s

)

a)

!1000

!800

!600

!400

!200

0

200

400

600

800

1000

Tim

e (s

)

b)

0 500 1000 1500 2000Distance along South-North array (km)

400600800100012001400160018002000

0

100

200

300

400

500

600

700

800


Tim

e (s

)

c)

400600800100012001400160018002000

0

100

200

300

400

500

600

700

800


Tim

e (s

)

d)

Figure 11. Passive surface-wave interferometry. (a) Cross-correlation result (red) compared with directly modeled re-sponse (black). The virtual source is indicated by the red

receiver in Figure 10a. The traces correspond to the irreg-ular receiver stations in the Western USA in Figure 10a.

(b) Green-shaded area: approximation of the point-spreadfunction along the South-North array. (c) MDD result (red)compared with directly modeled response (black). (d) As in

(c), but obtained from correlation and point-spread functiondefined by equations (59) and (60).

(the total duration of the modeled noise responses isapproximately two days).

Our aim is to turn one of the receivers of the South-North array (the one indicated by the red triangle) intoa virtual source and to retrieve the response of this vir-tual source at the irregular receiver stations in the West-ern USA. Following the crosscorrelation method we takethe noise response at the red receiver and crosscorrelateit with each of the responses at the receivers in the West-ern USA, i.e., we evaluate 〈u(xB , t) ∗ u(xA,−t)〉, withxA fixed at the red receiver and xB variable. The resultis shown in Figure 11a by the red traces. For compari-son, the black traces in this figure represent the directlymodeled surface-wave response of a source at the po-sition of the red receiver, using a source function S(t)equal to the average autocorrelation of the noise. Thereis a pronounced mismatch between the red and blackwaveforms, as a result of the irregularity of the sourcedistribution. Overall, the amplitudes of the direct waveare estimated too high, whereas the amplitudes of thescattered event are underestimated.

For the MDD approach we need to invert equa-tion (27), with, according to equation (31), the cor-relation function given by C(xB ,xA, t) = 〈u(xB , t) ∗uin(xA,−t)〉, and, according to equation (32), the point-spread function given by Γ(x,xA, t) = 〈uin(x, t) ∗uin(xA,−t)〉. These expressions contain the inwardpropagating waves at x and xA on Srec (which cor-responds with the South-North array). Because weconsider noise responses we cannot separate the in-ward propagating waves uin from the outward prop-agating waves uout by time-windowing. If we wouldhave multi-component receivers or two arrays close toeach other, we could apply decomposition, after whichC(xB ,xA, t) and Γ(x,xA, t) could be evaluated. Inthe following we assume that these conditions for de-composition are not fulfilled. Although separation bytime-windowing cannot be applied to the noise data,it can be applied to the correlation function. Recallthat Figure 11a represents the correlation of full wavefields, i.e., 〈u(xB , t) ∗ u(xA,−t)〉. From this we ex-tract 〈u(xB , t) ∗ uin(xA,−t)〉 by removing the eventsin the gray-shaded area as well as all events at neg-ative time (not shown in Figure 11a), which togetherrepresent 〈u(xB , t) ∗ uout(xA,−t)〉. Hence, we retainonly the green-shaded area in Figure 11a, represent-ing C(xB ,xA, t) = 〈u(xB , t) ∗ uin(xA,−t)〉. To obtainthe point-spread function, we first evaluate 〈u(x, t) ∗u(xA,−t)〉, with xA again fixed at the red receiverand x variable along the South-North array, see Fig-ure 11b. We remove the events in the white area, repre-senting 〈uout(x, t) ∗ uin(xA,−t)〉, and those in the gray-shaded area, representing 〈uin(x, t) ∗ uout(xA,−t)〉. Weretain the green-shaded area, representing 〈uin(x, t) ∗uin(xA,−t)〉+ 〈uout(x, t) ∗ uout(xA,−t)〉. Assuming theoutward propagating field is relatively small, we maythus approximate the events in the green-shaded area in


Figure 11b by Γ(x,xA, t) = 〈uin(x, t)∗uin(xA,−t)〉. Weare now ready to apply MDD, which involves inversionof equation (27), i.e., removing the effect of the point-spread function (the green-shaded area in Figure 11b)from the correlation function (the green-shaded area inFigure 11a). The result is shown by the red traces inFigure 11c, for display purposes convolved with S(t).Note that the wave forms of the direct wave as wellas those of the scattered wave match the directly mod-eled wave forms (the black traces) much better than inFigure 11a. We also observe some non-causal residuesat small travel times. These are mainly due to the factthat time-windowing is an imperfect method to sepa-rate events in the correlation functions. To reduce theamount of required time-windowing, we propose to re-place equations (31) and (32) by

C(xB ,xA, t) = 〈u(xB , t) ∗ u(xA,−t)〉, (59)

Γ(x,xA, t) = 〈uin(x, t) ∗ u(xA,−t)〉. (60)

Note that in both equations we made the same re-placement, i.e., we replaced uin(xA,−t) by u(xA,−t).Hence, equation (27) holds with the correlation andpoint-spread function defined by equations (59) and(60), respectively. The correlation function is obtainedfrom full wave fields, hence no time windowing is neededin Figure 11a. The point-spread function now consists of〈uin(x, t)∗uin(xA,−t)〉+〈uin(x, t)∗uout(xA,−t)〉, hence,only the events in the white area in Figure 11b need tobe removed to obtain an approximation of Γ(x,xA, t).Applying MDD by inverting equation (27), with the cor-relation and point-spread function defined by equations(59) and (60), respectively, we obtain the result shownby the red traces in Figure 11d (for display purposesconvolved with S(t)). Note that the match with the di-rectly modeled result is again very good and that thenon-causal residuals vanished almost completely.

6.2 Passive body-wave interferometry

Ambient noise is usually dominated by surface waves.Passive body-wave interferometry is only possible aftercareful suppression of surface waves (Draganov et al.2009).

Figure 12a shows the configuration for passivebody-wave interferometry by crosscorrelation. The sur-face Ssrc forms together with the Earth’s free surfacea closed boundary. Sources are only required on theopen surface Ssrc. The correlation function C(xB ,xA, t)involves the crosscorrelation of the full wave fields atxA and xB (Figure 12a only shows the direct path ofu(xA,xS , t) and the first reflected path of u(xB ,xS , t)).The crosscorrelation is integrated over the source posi-tions in case of transient sources (equation 11) or av-eraged over time in case of uncorrelated noise sources(equation 17). Equation (13) gives the relation betweenthe correlation function C(xB ,xA, t) and the Green’sfunction G(xB ,xA, t), which for the current configura-

AxBx

srcS

Sx!!! !!!

( , , )B AC tx x

( , , )B Su tx x ( , , )A Su tx x

a)

V

Bx recS

( )iSx

x!!! !!!

( , , )BG tx x

( )( , , )iB Sv tx x

b)

V

( )( , , )iSp tx x

Ax

Figure 12. Configurations for passive body-wave interfer-ometry. (a) In the crosscorrelation method the full wave-

fields u(xA,xS , t) and u(xB ,xS , t) are correlated and inte-

grated along the sources on Ssrc. The correlation functionC(xB ,xA, t) is proportional to G(xB ,xA, t)+G(xB ,xA,−t),

convolved with the autocorrelation of the sources. (b) Theconvolutional model used for interferometry by MDD. The

response v(xB ,x(i)S , t) − v(xB ,x

(i)S , t) is equal to the convo-

lution of the Green’s function G(xB ,x, t) and the reference

wave field p(x,x(i)S , t), integrated along the receivers at Srec.

tion should be interpreted as the reflection response ofthe subsurface. It includes primary and multiple reflec-tions (internal multiples and surface-related multiples)as well as a direct wave. Equations (11), (13) and (17)have been derived for the situation that the receiversxA and xB are in V. In that case u stands for acous-tic pressure (p) and G(xB ,xA, t) represents the acousticpressure at xB due to an impulsive point source of vol-ume injection rate at source position xA. The equationsremain valid when the receivers xA and xB are at thefree surface. In that case u stands for the normal particlevelocity (v) and G(xB ,xA, t) represents the normal par-ticle velocity at xB due to an impulsive normal tractionsource at xA (Wapenaar & Fokkema 2006).

We illustrate this method with a numerical exam-ple. Consider the configuration in Figure 13a, whichconsists of a horizontally layered target below a ho-mogeneous overburden. The green triangles at the freesurface denote 51 regularly spaced vertical geophoneswith spacing ∆x1 = 40 m (only nine geophones areshown). The blue dots below the layered target denote250 irregularly spaced uncorrelated noise sources withan average lateral spacing of ∆x1 = 20 m and a cen-tral frequency of 22 Hz. Note the clustering of sourcesaround x1 ≈ −1400 m and x1 ≈ 500 m. The responsesof these sources are shown in Figure 13b (only 3 s of 25minutes of noise is shown). We denote these responses

170 Wapenaar et al.

1 … 26 … 51

1500

c (m/s)

2000

1950 1900 1800 2200

1200

(kg/m3)

1500

0

Receiver0

1500 1450 1400 1600

-2000 -1000 0 1000 2000x1 (m)

a)

x 3(m

)

0

500

1000

1500

2000

2500

!800 !400 0 400 8000

0.5

1

1.5

2

2.5

3

Offset (m)

Tim

e (s

)

b)

!800 !400 0 400 800!1.5

!1

!0.5

0

0.5

1

1.5

Offset (m)

Tim

e (s

)

c)

!800 !400 0 400 8000.8

0.9

1

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1.2

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e (s

)

d)

!800 !400 0 400 8000.8

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)

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!800 !400 0 400 800!1.5

!1

!0.5

0

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e (s

)

f)

Figure 13. Numerical example of ambient-noise body-waveinterferometry. (a) Configuration with a horizontally lay-

ered target below a homogeneous overburden and a free sur-

face. There is an irregular distribution of uncorrelated noisesources below the target. (b) Modeled noise response, ob-

served by receivers at the free surface. (c) Correlation func-

tion Cv,v(xB ,xA, t) = 〈v(xB , t) ∗ v(xA,−t)〉 (fixed xA, vari-able xB), clipped at 20 % of the maximum amplitude. (d)

Zoomed version of the causal part of the correlation function

(red traces), compared with the directly modeled scatteredpart of the Green’s function, Gs(xB ,xA, t) ∗ S(t). (e) Idem,

for the MDD result. This was obtained by deconvolving thecausal events in (c) by the point-spread function (the strong

event in (c) around t = 0). (f) Directly modeled correlation

function Cvs,vs (xB ,xA, t) (same amplitude scaling as in c).This weak event is ignored in the MDD method.

as v(xA, t) and v(xB , t), where xA and xB can be any ofthe 51 geophones. We evaluate the correlation function

Cv,v(xB ,xA, t) = 〈v(xB , t) ∗ v(xA,−t)〉, (61)

averaged over 25 minutes of noise. The result is shownin Figure 13c for fixed xA (geophone number 26) andvariable xB (geophones 1 − 51). According to equa-tion (13) this correlation function is proportional to{G(xB ,xA, t)+G(xB ,xA,−t)}∗S(t). Figure 13d showsa zoomed version of the causal part of the correlationfunction Cv,v(xB ,xA, t) (the red traces) and the directlymodeled reflection response Gs(xB ,xA, t)∗S(t) with thesource xA at the position of geophone 26 (the blacktraces). Here Gs(xB ,xA, t) is the scattered Green’s func-tion, i.e., the total Green’s function G(xB ,xA, t) mi-nus the direct wave. Note that the arrival times of thecorrelation function nicely match those of the directlymodeled reflection response, but the wave forms andamplitudes are not accurately reconstructed. This dis-crepancy is a consequence of the irregular source distri-bution, in particular the clustering around x1 ≈ −1400m and x1 ≈ 500 m (Figure 13a), which implies that theintegral along Ssrc is not properly discretized.

Next we discuss passive reflected-wave interferom-etry by MDD. Figure 12b shows the modified configu-ration. This configuration is different from other MDDconfigurations discussed in this paper in the sense thatthe source positions x

(i)S are now in V. Hence, because

this configuration is not a special case of that in Figure1b we cannot make use of equation (26) without makingsome modifications. We previously dealt with the con-figuration of Figure 12b (Wapenaar et al. 2008b) andderived the following convolution-type representation

v(xB ,x(i)S , t)− v(xB ,x

(i)S , t) =Z

SrecG(xB ,x, t) ∗ p(x,x

(i)S , t) dx. (62)

Here the bars denote a reference situation with absorb-ing boundary conditions at Srec. Hence, p(x,x

(i)S , t) and

v(xB ,x(i)S , t) are the pressure and normal particle veloc-

ity that would be measured at Srec in absence of the freesurface. The Green’s function G(xB ,x, t) in equation(62) represents again the normal particle velocity at xBdue to an impulsive normal traction source at source po-sition x, both at Srec. It is defined in the actual mediumwith the free-surface boundary condition at Srec. Forlater convenience we subtract the direct wave contribu-tion from both sides of equation (62), yielding

vs(xB ,x(i)S , t) =

ZSrec

Gs(xB ,x, t) ∗ p(x,x(i)S , t) dx, (63)

where Gs(xB ,x, t) is again the scattered Green’s func-tion (i.e., the total Green’s function minus the directwave) and where

vs(xB ,x(i)S , t) = v(xB ,x

(i)S , t)− 2v(xB ,x

(i)S , t). (64)


The factor 2 reflects the fact that the direct wave inv(xB ,x

(i)S , t) is twice the direct wave v(xB ,x

(i)S , t) in

the situation without free surface. Note that, despite thedifferences with equation (26), the form of equation (63)is the same as that of equation (26). Hence, analogousto equation (27) we may write

Cvs,v(xB ,xA, t) =

ZSrec

Gs(xB ,x, t) ∗ Γ(x,xA, t) dx,

(65)where the correlation function and the point-spreadfunction are defined as

Cvs,v(xB ,xA, t) =Xi

vs(xB ,x(i)S , t) ∗ v(xA,x

(i)S ,−t),

(66)

Γ(x,xA, t) =Xi

p(x,x(i)S , t) ∗ v(xA,x

(i)S ,−t) (67)

in the case of transient sources, or as

Cvs,v(xB ,xA, t) = 〈vs(xB , t) ∗ v(xA,−t)〉, (68)

Γ(x,xA, t) = 〈p(x, t) ∗ v(xA,−t)〉 (69)

in the case of uncorrelated noise sources. A complica-tion is that these correlation functions contain fieldsin a reference situation with an absorbing boundary atSrec. To see how this can be dealt with for transientsources and for noise sources, consider again the config-uration of Figure 13a. When the sources are transients,the first arrivals (including internal multiple scattering

in the target) in v(xA,x(i)S , t) are well separated in time

from the surface-related multiples, so the reference re-sponse v(xA,x

(i)S , t) can be extracted by applying a time

window to v(xA,x(i)S , t) and multiplying the result by

one-half. Both v and p are outward-propagating wavesat the absorbing boundary Srec, so p can be obtainedfrom v using the equation of motion and the one-waywave equation for outward-propagating waves, accord-ing to

−jωρˆv = ni∂i ˆp = −jH1 ˆp. (70)

Because of the near-vertical incidence of the waves atSrec, in practice it suffices to make the following approx-imation

p(x,x(i)S , t) ≈ ρ1c1v(x,x

(i)S , t), (71)

where ρ1 and c1 are the mass density and propagationvelocity, respectively, directly below Srec. This gives allfields needed in equations (64), (66) and (67) to evaluatethe correlation function and the point-spread function,after which Gs(xB ,x, t) can be resolved by MDD fromequation (65), see Wapenaar et al. (2008b) for a numer-ical example.

When the sources are noise sources we cannot sep-arate the reference fields from the measured fields bytime windowing, but van der Neut et al. (2010) proposeto make such a separation in the correlation function,as follows. Using v = 2v + vs (equation 64) we write for

the correlation function, as defined in equation (61),

Cv,v(xB ,xA, t) = (72)

{4Cv,v + 2Cvs,v + 2Cv,vs + Cvs,vs}(xB ,xA, t),

with Cvs,v(xB ,xA, t) defined in equation (68), and

Cv,v(xB ,xA, t) = 〈v(xB , t) ∗ v(xA,−t)〉, (73)

Cv,vs(xB ,xA, t) = 〈v(xB , t) ∗ vs(xA,−t)〉, (74)

Cvs,vs(xB ,xA, t) = 〈vs(xB , t) ∗ vs(xA,−t)〉. (75)

For the configuration of Figure 13a, the total correla-tion function Cv,v(xB ,xA, t) is shown in Figure 13c (forfixed xA and variable xB). Assuming Cvs,vs(xB ,xA, t)is weak, the other three terms can be easily obtainedfrom Figure 13c. The strong event between the twodashed lines is the correlation of the reference response,4Cv,v(xB ,xA, t). The causal event (below the lowerdashed line) is 2Cvs,v(xB ,xA, t) and the acausal event(above the upper dashed line) is 2Cv,vs(xB ,xA, t). Toinvert equation (65), we also need Γ(x,xA, t), definedin equation (69). Analogous to equation (70), this canbe obtained from

−jωρCv,v(x,xA, t) = −jH1Γ(x,xA, t). (76)

Assuming again near-vertical incidence, this can be ap-proximated by

Γ(x,xA, t) ≈ ρ1c1Cv,v(x,xA, t). (77)

Having determined these functions, Gs(xB ,x, t) canagain be resolved from equation (65) by MDD. The re-sult (for x = xA fixed at the central geophone and vari-able xB) is shown in Figure 13e (red traces), where it iscompared with the directly modeled scattered Green’sfunction (black traces). Both results have been con-volved with S(t) to facilitate the comparison with thecorrelation of Figure 13d. The MDD result in Figure13e matches the directly modeled response significantlybetter than the correlation in Figure 13d.

Note that we assumed that Cvs,vs(xB ,xA, t) isweak. Figure 13f shows this function obtained from di-rect modeling and displayed with the same amplitudescaling as Cv,v(xB ,xA, t) in Figure 13c. We observe thatfor this example it is indeed weak. The main contribu-tion is concentrated around t = 0, so it has only a smalleffect on the estimated point-spread function, which isobtained (via equations (73) and (77)) from the eventbetween the dashed lines in Figure 13c. When there arestrong reflectors in the subsurface, Cvs,vs(xB ,xA, t) maynot be weak and distort the estimation of the point-spread function.

An alternative deconvolution approach for pas-sive reflected-wave interferometry is presented by vanGroenestijn & Verschuur (2010). Their method does notuse the separation of events in the correlation function,but is adaptive and aims to minimize the energy in theestimated reference wave field v(xA, t). Since the reflec-tions in the total noise field v(xA, t) may interfere con-

172 Wapenaar et al.

structively as well as destructively with the referencenoise field v(xA, t), there is no physical justification thatthe energy in v(xA, t) should be minimized. It remainsan open question which of the approaches functions bestin case of strongly scattering media.

7 DISCUSSION AND CONCLUSIONS

The methodology of seismic interferometry (or Green’sfunction retrieval) by crosscorrelation has a number ofattractive properties as well as several limitations. Themain attractiveness of the method is that a determinis-tic medium response can be obtained from controlled-source or passive noise measurements, without the needto know the medium parameters nor the positions ofthe sources. The fact that this deterministic responseis obtained by a straightforward crosscorrelation of tworeceiver responses has also contributed to the popular-ity of the method. The main underlying assumptionsare that the medium is lossless and that the wave fieldis equipartitioned, meaning that the net power-flux is(close to) zero (Malcolm et al. 2004; Sanchez-Sesmaet al. 2006). It has been shown by Fan & Snieder (2009)that equipartitioning is a necessary but not a sufficientcondition.

For open systems the method works well when thereceivers that are used in the correlation process are sur-rounded by a regular distribution of independent tran-sient or uncorrelated noise sources with equal autocor-relation functions. However, in practical situations themethod may suffer from irregularities in the source dis-tribution, asymmetric illumination, intrinsic losses, etc.

Figures 3a and 3b illustrate configurations with anirregular source distribution in a finite region, illumi-nating the medium from one side only. For these con-figurations, equation (27) shows that the crosscorrela-tion function C(xB ,xA, t) is a distorted version of thesought Green’s function Gd(xB ,x, t). The distortion isquantified by the interferometric point-spread functionΓ(x,xA, t), which blurs the source of the Green’s func-tion in the spatial directions and generates ghosts in thetemporal direction (van der Neut & Thorbecke 2009).As such, this interferometric point-spread function playsa similar role as the point-spread function in optical,acoustic and seismic imaging systems (Born & Wolf1965; Norton 1992; Schuster & Hu 2000). When thepoint-spread function of an imaging system is known,the resolution of an image can be improved by decon-volving for the point-spread function (Jansson 1997; Huet al. 2001). For example, early images of the HubbleSpace Telescope were blurred by a flaw in the mirror.These images were sharpened by deconvolving for thepoint-spread function of the flawed mirror (Boden et al.1996). In a similar way, the blurred source with ghosts ofthe Green’s function obtained by crosscorrelation can bedeblurred and deghosted by deconvolving for the inter-ferometric point-spread function, i.e., by inverting equa-

tion (27) by MDD. An interesting aspect is that theinterferometric point-spread function can be obtaineddirectly from measured responses, hence, it does notrequire knowledge of the complex medium nor of thesources. This means that it automatically accounts forthe distorting effects of the irregularity of the sources(including variations in their autocorrelation functions),and of the medium inhomogeneities, including multiplescattering.

The MDD method has, of course, also its limita-tions. First, in order to retrieve the Green’s functionGd(xB ,x, t) it does not suffice to have two receiversonly, at x and xB , because x is assumed to be an el-ement of a regular (or regularized) array of receiversalong Srec, see Figure 3. Second, the expressions for thecorrelation function (equations 28 and 31) and for thepoint-spread function (equations 29 and 32) contain theinward-propagating field uin(x, t) on Srec (propagatinginto V). In order to extract this inward-propagating fieldfrom the total field, either multi-component receivers ortwo receiver surfaces close to each other are required.Alternatively, when the scattering is weak, one couldcorrelate the total fields and extract the point-spreadfunction by applying a time window around t = 0.Third, MDD involves an inversion of an integral equa-tion, which in practice is achieved by matrix inver-sion. This matrix inversion can be unstable. The well-posedness of this inverse problem depends on the num-ber of available sources, the source aperture, the band-width, and, for multi-component data, on the number ofindependent source components. In practical situationsa spectral analysis of the point-spread function helps toassess for which spatial and temporal frequencies theinversion can be carried out in a stable sense (van derNeut et al. 2011). Finally, because of the matrix inver-sion, the costs of the MDD method are higher than thoseof the correlation method. For the 2D examples in thispaper, the costs of the matrix inversions were moderate,despite the fact that we used direct solvers only. For 3Dapplications we expect that MDD is still feasible, par-ticularly when use is made of iterative solvers.

Despite the mentioned limitations, for applicationsin which the data are measured with arrays of receivers,Green’s function retrieval by MDD has the potentialto obtain virtual sources that are better focused andhave less distortions by spurious multiples than thoseobtained by crosscorrelation.

ACKNOWLEDGEMENTS

This work is supported by the Netherlands ResearchCentre for Integrated Solid Earth Science (ISES),The Netherlands Organisation for Scientific Research(NWO, Toptalent 2006 AB), and the Dutch TechnologyFoundation STW (grants VENI.08115 and DCB.7913).We thank associate editor Andrew Curtis and two


anonymous reviewers for their constructive commentsand suggestions, which helped to improve this paper.

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APPENDIX A: ANALYTICALEVALUATION OF THE POINT-SPREADFUNCTION

We evaluate the interferometric point-spread function,as defined in equation (29), for the situation of a homo-geneous lossless medium (propagation velocity c) and aregular distribution of sources with equal autocorrela-tion functions S(t) along an open source boundary. Forour analysis we rewrite the sum over the source posi-tions x

(i)S as an integral along the source boundary Ssrc,

hence

Γ(x,xA, t) =

ZSsrc

uin(x,xS , t) ∗ uin(xA,xS ,−t) dxS ,

(A− 1)with x and xA at the receiver boundary Srec. Note that,unlike in equation (11), the source boundary Ssrc is notenclosing the receivers, hence, Γ(x,xA, t) does not con-verge to(ρc/2){G(x,xA, t) + G(x,xA,−t)} ∗ S(t). For the fol-lowing analysis we let Ssrc and Srec be two horizon-tal boundaries, with Srec a distance ∆x3 below Ssrc.For convenience we consider the 2D situation, so theboundary integral becomes a line integral along x1,S .For the chosen configuration the fields under the inte-gral are shift-invariant, i.e., they are only functions ofthe relative distances x1 − x1,S and x1,A− x1,S , respec-tively. Similarly, the point-spread function is a functionof x1 − x1,A only. Choosing x1,A = 0, equation (A-1)can thus be rewritten as

Γ(x1, t) =R ∞−∞ uin(x1 − x1,S , ∆x3, t) ∗ uin(−x1,S , ∆x3,−t) dx1,S ,

(A− 2)where Γ(x1, t) is a short notation for Γ(x1, 0, t), etc. Wedefine the double Fourier transform of a space- and time-dependent function u(x1, t) as

u(k1, ω) =

Z ∞

−∞

Z ∞

−∞exp{−j(ωt− k1x1)}u(x1, t) dtdx1.

(A− 3)Hence, equation (A-2) transforms to

Γ(k1, ω) = uin(k1, ∆x3, ω){uin(k1, ∆x3, ω)}∗. (A− 4)

In section 4.2 we argued that the representation forMDD (equation (26)) holds, irrespective of the typeof sources at Ssrc. For convenience of the analysis weconsider dipole sources emitting a signal s(t) (this isfor example representative for the situation of verticallyoriented vibrators at a free surface). The expression for


uin(k1, ∆x3, ω) is thus given by

uin(k1, ∆x3, ω) = s(ω)×

8>>>><>>>>:exp{−j sgn(ω)

pk2 − k2

1 ∆x3}for k2

1 ≤ k2

exp{−p

k21 − k2 ∆x3}

for k21 > k2,

(A− 5)with k = ω/c. Substitution into equation (A-4) gives

Γ(k1, ω) = S(ω)×

8>><>>:1 for k2

1 ≤ k2

exp{−2p

k21 − k2 ∆x3}

for k21 > k2,

(A− 6)

with S(ω) = |s(ω)|2. Hence, for propagating waves(k2

1 ≤ k2) the Fourier-transformed point-spread func-tion Γ(k1, ω) is equal to the power spectrum S(ω)of the dipole sources, whereas for evanescent waves(k2

1 > k2) it is exponentially decaying. Note thatfor the limit ∆x3 → 0 the point-spread function ap-proaches S(ω) for all k1. Hence, for this case we havelim∆x3→0 Γ(x1, t) = {δ(x1)δ(t)}∗S(t), meaning that theonly band-limitation is caused by the autocorrelationof the sources. However, for any realistic value of ∆x3

the exponentially decaying character of the evanescentwaves imposes a more serious band-limitation to thedelta function. Already for ∆x3 larger than a few wave-lengths it is reasonable to approximate the exponen-tially decaying term in equation (A-6) by zero for allk21 > k2, meaning that for a given frequency ω the pass-

band is −|ω|/c ≤ k1 ≤ |ω|/c. Hence, the inverse trans-formation to the space-frequency domain is obtained as

Γ(x1, ω) =1

2π

Z ∞

−∞Γ(k1, ω) exp(−jk1x1) dk1

=S(ω)

2π

Z |ω|/c

−|ω|/cexp(−jk1x1) dk1

=sin(|ω|x1/c)

πx1S(ω). (A− 7)

Note that this expression is identical to the resolu-tion function for 2D seismic migration (Berkhout 1984).When the source array is finite, the pass-band is furtherreduced to−|ω| sin αmax/c ≤ k1 ≤ |ω| sin αmax/c, where αmax isthe maximum propagation angle. Hence, for this situa-tion we replace the velocity c in equation (A-7) by theapparent velocity ca = c/ sin αmax.

To obtain the space-time domain expression we firstrewrite equation (A-7) as

Γ(x1, ω) = −j sgn(ω)j sin(ωx1/ca)

πx1S(ω) (A− 8)

= −j sgn(ω)exp(jωx1/ca)− exp(−jωx1/ca)

2πx1S(ω).

This product of three functions in the frequency domaintransforms to a convolution of the corresponding time-

!480 !240 0 240 480!1

!0.5

0

0.5

1


Tim

e (s

)

Figure A1. The point-spread function Γ(x1, t) (clipped at

20% of its maximum amplitude) for the situation of a ho-mogeneous lossless medium (propagation velocity 2000 m/s)

and an infinite regular source array. The autocorrelation of

the sources is a Ricker wavelet with a central frequency of 20Hz.

domain functions, as follows

Γ(x1, t) =1

πt∗ δ(t + x1/ca)− δ(t− x1/ca)

2πx1∗ S(t).

(A− 9)For x1 6= 0 this gives

Γ(x1, t) =1

2π2x1

»1

t + x1/ca− 1

t− x1/ca

–∗ S(t),

(A− 10)and for x1 → 0

Γ(0, t) =1

π2cat∗ ∂S(t)

∂t. (A− 11)

Γ(x1, t) is illustrated in Figure A1 for c = 2000 m/s,αmax = π/2 (corresponding to an infinite source array)and S(t) a Ricker wavelet with a central frequency of20 Hz. Note that Γ(x1, t) reveals an X-pattern with themaximum amplitude at (x1, t) = (0, 0) and decayingamplitudes along lines t = ±x1/c. The point-spreadfunction Γ(x1, t) can be seen as a band-limited deltafunction δ(x1)δ(t), with the temporal band-limit im-posed by the autocorrelation function S(t) and the spa-tial band-limit imposed by the evanescent waves and,for a finite source array, the absence of large propaga-tion angles.

For the derivation in this appendix we considered a2D configuration and assumed the sources are dipoles.For other situations (3D configuration and/or monopolesources) the details are different, but in all cases the an-alytical point-spread function reveals the characteristicX-pattern (or a two-sided conical pattern in 3D).

178 Wapenaar et al.

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