GEOPHYSICS, VOL. 66, NO. 1 (JANUARY-FEBRUARY 2001); P. 327–341, 6 FIGS., 1 TABLE.

Elimination of free-surface related multipleswithout need of the source wavelet

Lasse Amundsen∗

ABSTRACT

This paper presents a new, wave-equation basedmethod for eliminating the effect of the free surface frommarine seismic data without destroying primary ampli-tudes and without any knowledge of the subsurface.Compared with previously published methods which re-quire an estimate of the source wavelet, the presentmethod has the following characteristics: it does not re-quire any information about the marine source arrayand its signature, it does not rely on removal of the di-rect wave from the data, and it does not require anyexplicit deghosting. Moreover, the effect of the sourcesignature is removed from the data in the multiple elimi-nation process by deterministic signature deconvolution,replacing the original source signature radiated from themarine source array with any desired wavelet (within thedata frequency-band) radiated from a monopole pointsource.

The fundamental constraint of the new method is thatthe vertical derivative of the pressure or the vertical com-

ponent of the particle velocity is input to the free-surfacedemultiple process along with pressure recordings. Theseadditional data are routinely recorded in ocean-bottomseismic surveys. The method can be applied to conven-tional towed streamer pressure data recorded in the wa-ter column at a depth which is greater than the depthof the source array only when the pressure derivativecan be estimated, or even better, is measured. Sincethe direct wave and its source ghost is part of the free-surface demultiple, designature process, the direct ar-rival must be properly measured for the method to worksuccessfully.

In the case when the geology is close to horizontallylayering, the free-surface multiple elimination methodgreatly simplifies, reducing to a well-known determinis-tic deconvolution process which can be applied to com-mon shot gathers (or common receiver gathers or com-mon midpoint gathers when source array variations arenegligible) in the τ -p domain or frequency-wavenumberdomain.

INTRODUCTION

Multiple reflections in marine seismic data are an old andlong-standing problem in exploration seismology. If multiplesare not properly attenuated, they can be misinterpreted as pri-mary reflections, or interfere and overlap with primaries. Manyprocessing algorithms are based on the fundamental assump-tion that seismic data contain primaries only. For this reasonalso, effective demultiple schemes are required.

Even though many methods have been developed for sup-pressing multiples, they are based on theoretical models andprerequisites that are not always met. Each method is based ondifferent principles and has different strengths and weaknesses.As remarked by Weglein (1999a), this problem motivates thesearch for new demultiple concepts and algorithms to add to

Manuscript received by the Editor March 12, 1999; revised manuscript received March 2, 2000.∗Statoil Research Centre, Postuttak, N-7005 Trondheim, Norway, and The Norwegian University of Science and Technology, Department of Physics,N-7491 Trondheim, Norway. E-mail: lam@statoil.com.c© 2001 Society of Exploration Geophysicists. All rights reserved.

our toolbox of methods. Demultiple methods which have thepotential and promise of attacking obstinate multiples whilepreserving primaries should be considered even though theymay place extra demands on complete wavefield recording inthe seismic experiment.

During the last two decades, a considerable effort has beendevoted to the problem of optimally attenuating all multi-ples related to the sea surface, which is a perfect reflectorfor all upgoing energy. Surface multiples are events that haveat least one downward reflection from the sea surface. Onewave-equation based approach that has received much atten-tion is to transform the recorded data during processing tonew, desired data which would be recorded in a hypotheticalexperiment with no sea surface present. Contributions in thisarea of research and algorithm development are Kennett

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328 Amundsen

(1979), Berkhout (1982, 1999), Verschuur et al. (1989, 1992),Fokkema and van den Berg (1990, 1993), Carvalho et al. (1991,1992, 1994), Verschuur (1991), van Borselen et al. (1992, 1996),Dragoset and MacKay (1993), Araujo et al. (1994), Matson andWeglein (1996), Ikelle (1997a, b, 1998, 1999a, b), Ikelle andJaeger (1997), Weglein et al. (1997), Holvik and Amundsen(1998), Ziolkowski et al. (1998, 1999), and Weglein (1999a, b).This class of wave-equation demultiple approach has been for-mulated to process pressure recordings from conventional ma-rine surveys with pressure-sensitive hydrophones deployed ina towed streamer. According to theory, the demultiple tech-nique eliminates the free-surface-related multiples while pre-serving primary amplitudes. No assumptions about the earthbelow the receivers are made. The demultiple theory, how-ever, is subject to some constraints. The most important con-straint is that source signature information is needed for thefree-surface demultiple process. In most practical implemen-tations of this method, the source signature is estimated dur-ing the demultiple processing by ad-hoc postulating that therequired signature is the one which minimizes the energy inthe demultipled field. Another constraint related to the prac-tical implementation of the demultiple theory is that the di-rect wave (including the sea-surface ghost) must be removedfrom the data. Only the reflected part of the data is input todemultiple processing. In deep water areas, the direct wavecan be muted, and thus represents no problem. In most cases,however, the direct wave interferes with the reflected data.When the source signature is known, one may model the di-rect wave for subsequent subtraction to try to overcome thisconstraint. A third constraint of a practical nature is that thepressure data should be deghosted before demultiple pro-cessing. One approximative but workable solution is to in-clude the ghost factors in the source signature. Any estima-tion scheme for the source signature then includes the ghosteffects.

In this paper, I present a new method to eliminate the ef-fect of the sea surface from marine seismic data recorded at adepth level below the depth level of the source. The recordingdepth assumption is not a severe restriction because most ofthe seismic data are acquired in this source-receiver geometry.The characteristics of the method are that it does not requireany information about the source array and its signature, itdoes not rely on removal of the direct wave from the data,and it does not require any explicit deghosting. Furthermore,implicit in the algorithm is a source signature deconvolutionprocess which, according to theory, transforms the signature ofthe source array into any desired monopole point-source sig-nature. The fundamental constraint of this method, however,is that the vertical derivative of the pressure or the verticalcomponent of the particle velocity is input to the free-surfacedemultiple process along with pressure recordings. These ad-ditional data are routinely recorded in ocean-bottom seismicsurveys which deploy hydrophones on the sea floor to recordthe pressure field and geophones to record the particle ve-locity field. Hence, the proposed method is ideally designedfor a free-surface demultipling of split-spread ocean-bottomseismic dual-sensor data. Another data acquisition techniquethat gives the vertical derivative of the pressure or the verti-cal component of the particle velocity is data recording withdual streamers where the pressure field is recorded at two dif-

ferent depth levels (Sønneland et al., 1986; Amundsen, 1993;Amundsen et al., 1995). For free-surface demultipling of con-ventionally towed streamer data, the best solution to the de-multiple problem is obviously to develop technology to recordthe vertical derivative of the pressure together with the pres-sure field. If the new demultiple technique reaches its esti-mated promise, I propose that the seismic acquisition methodbe extended to record dual-wave data. Finally, I note that whenthe source signature is known, for instance from near-fieldrecordings, the vertical derivative of the pressure field can beestimated from the pressure data in the recording geometrywith streamer below source depth level (Corrigan et al., 1991;Amundsen et al., 1995b). It remains to be investigated whetherit is better to estimate the vertical derivative of the pressurefield using the source wavelet to avoid the requirement of thesource wavelet in the demultiple algorithm, or to demultiplepressure data with an estimate of the wavelet as is currentlydone.

To my knowledge, only one free-surface demultiple methodvalid for any inhomogeneous earth has been proposed that usesthe additional recording of the vertical particle velocity at thesea floor. However, this method (as published by Ziolkowskiet al., 1998, 1999) still suffers from the constraints that relate topreviously published schemes: it requires a source signature es-timate, and it requires that the direct wave be subtracted fromthe data. This latter prerequisite is a nontrivial issue, particu-larly for ocean-bottom seismic data. The free-surface demul-tiple method published in the present paper and its extensionto free-surface demultipling of marine multicomponent ocean-bottom seismic data has been presented in conference proceed-ings (Amundsen, 1999a, b; Amundsen and Ikelle, 2000). Fur-ther, Amundsen et al. (2000b) presented an application of thismethod to an ocean-bottom seismic dataset acquired offshoreNorway.

Several demultiple schemes which do not aim at eliminat-ing all free-surface related multiples from ocean-bottom seis-mic data have been developed. These methods are based onup/down splitting of the recorded wavefield just below the seabottom, and removing all water-layer reverberations (down-going energy just below the sea floor) from the data. For moreinformation about these methods, see White (1965), Barr andSanders (1989), Amundsen et al. (1995a, 2000a), Osen et al.(1996, 1998, 1999a, b), Schalkwijk et al. (1997), and Holviket al. (1999).

The paper is organized as follows. First, I discuss themotivation for this work. For a horizontally layered medium, ithas long been realized that the free-surface effect can be elim-inated from the pressure recording when the field is recordedbelow the source depth level by dividing the upgoing by thedowngoing pressure component. After some discussion of this,I show how this well-known result can be generalized to de-velop a scheme that is valid for any inhomogeneous subsurface.I find that an integral equation governs the relationship be-tween the the desired demultipled and designatured pressurefield and the recorded pressure field. This integral equationfollows from Rayleigh’s reciprocity theorem (which is brieflyreviewed in Appendix A). Finally, I give a numerical examplewhich illustrates the free-surface demultiple and designatureprinciple applied to data modeled above a horizontally layeredmedium.

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Elimination of Free-Surface Multiples 329

MOTIVATION FOR THE PRESENT WORK: OLD RESULTSFOR A HORIZONTALLY LAYERED EARTH

It has long been recognized that for a horizontally layeredmedium the reflection response in the hypothetical seismic ex-periment with the free surface absent can be estimated from thereflection response in the physical seismic experiment with thefree surface present. Because of the horizontal layering,the problem is most conveniently studied in the frequency-wavenumber domain. In this domain, let P be the transformof the pressure recording p in the physical seismic experiment,and let P be the transform of the pressure p that would berecorded in the hypothetical seismic experiment; p is the de-sired pressure field excited by a monopole point source withdesired source wavelet a= a(ω), where ω is the circular fre-quency. Furthermore, assume that the fields can be separatedinto direct wave contributions pd and pd from source to re-ceivers and reflected contributions pr and pr from the stack oflayers according to

p = pd + pr ; P = Pd + Pr , (1)

p = pd + pr ; P = Pd + Pr . (2)

Note that here the direct wave pd is defined to include its seasurface ghost. Kennett (1979) devised a scheme which calcu-lates by an algebraic transformation the reflected part of thedesired field Pr from the reflected part of the recorded field Pr

with the assumption that the source in the physical experimentis a point source with a known wavelet. Source and receiverdeghosting is part of this free-surface demultiple process. Thephysical source and receivers can be located anywhere in thewater column. However, it has also long been realized thatwhen the pressure recording can be split into upgoing anddowngoing constituents (u and d, respectively) such that

p = u+ d; P = U + D, (3)

and, in addition, when the pressure is recorded anywhere belowthe depth level of the physical source (which may consist ofan array of guns), then the reflected part of the desired fieldcan be found by spectral division of the transformed upgoingand downgoing components according to (see, e.g., Amundsen,1993)

Pr = − a

2ikz

U

D, (4)

where kz=√

k2− k2x − k2

y is the vertical wavenumber, k=ω/c isthe wavenumber, c is the wave propagation velocity in water,and kx and ky are the horizontal wavenumbers. Likewise, thedesired total pressure can be found from equation (4) accordingto

P = − a

2ikz

P

D(5)

by realizing that Pd =−a/(2ikz). Note that factor −(2ikz)−1 isrequired to take into account the monopole point source effectof the hypothetical experiment (implying that in a homoge-neous medium the wavefield from the desired source is spheri-cal). This monopole point source effect, however, can automat-ically be accounted for by deconvolving U or P with −(2iωρ)−1

times the downgoing constituent D(Vz) of the vertical compo-nent of the particle velocity, which is related to downgoing

pressure constituent D according to [see Appendix B, equa-tions (B-9) and (B-11)]

D(Vz) ≡ kz

ρωD, (6)

where ρ is the density of water. For instance, the free-surfacedemultiple process (5) then is expressed by

P = − a

2iωρ

P

D(Vz). (7)

Noticing that multiplication in frequency domain by − iω cor-responds to temporal differentiation, we observe that the spik-ing deconvolution operator which eliminates all free-surface-related multiples actually is proportional to the time derivativeof the downgoing part of the vertical particle velocity field (cf.Amundsen, 1999b, and Amundsen and Ikelle, 2000).

One way to obtain the required upgoing and downgoingwaves is to record (or estimate) the vertical component of theparticle velocity vz in addition to the pressure recording p. Thevertical derivative of the pressure is related to vz through theequation of motion (A-1). Let Vz be the Fourier transform ofvz. As briefly discussed in Appendix B, the components U , D,and D(Vz) can be computed from P and Vz as

U = 12

(P − ωρ

kzVz

), (8)

D = P −U = 12

(P + ωρ

kzVz

), (9)

and

D(Vz) = 12

(Vz+ kz

ρωP

). (10)

Observe two attractive merits of equations (4), (5), and (7).First, no information about the marine source array and itswavlet in the physical experiment is required for the removalof the free-surface related multiples. In fact, since the effectof the source wavelet is contained both in U and P and in Dand D(Vz), it is canceled by the spectal division process. Implicitin the free-surface multiple elimination schemes (4), (5), and(7) is thus a source signature deconvolution process. The de-sired wavelet a is applied in a user-defined fashion to band-limit the demultipled data. By choosing a= 1, the free-surfacedemultipled and designatured response is the Green’s functionresponse of the hypothetical seismic experiment with the free-surface absent. Second, the direct wave pd is not subtractedfrom the input data to the free-surface demultiple process. Infact, the direct arrival pd is an important constituent of thedata. Its proper recording allows the designature process towork successfully, and further ensures correct prediction ofamplitude and phase of the free-surface-related multiples tobe eliminated. A possible drawback of schemes (4), (5), and(7) is that spectral division may be unstable.

The central argument of this paper is that source waveletinformation from the physical experiment is not required forfree-surface pressure demultipling when the particle velocitycomponent is measured or estimated. This conclusion can alsobe reached by another chain of reasoning. The free surface de-multipling of pressure data (as formulated, e.g., by Kennett,1979) requires two sets of data: the pressure record itself andthe source wavelet. However, it is well known that when the

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330 Amundsen

receiver level is below the source level then there exists a “trian-gle relationship” between the pressure, the vertical componentof particle velocity, and the source wavelet. When any of twoquantities are known, the third can be predicted from theory(see, e.g., Weglein and Secrest, 1990; Amundsen et al. 1995).This triangle relationship is briefly discussed in Appendix B[see equation (B-20)]. Thus, present-day free-surface demulti-ple theory can be reformulated to depend on the vertical parti-cle velocity instead of the source wavelet. Likewise, a trianglerelationship exists between the source wavelet and upgoingand downgoing waves recorded below the source level [seeequation (B-22)], and a triangle relationship governs the rela-tion between the pressure field and its upgoing and downgoingcomponents, simply p= u+ d. Therefore, it follows that to-day’s free-surface demultiple algorithms can be reformulatedto demultiple pressure data from the upgoing and downgoingfields.

The triangle relationship among the pressure field, the ver-tical component of particle velocity, and the source waveletis a powerful relationship. It tells us that any process that re-quires pressure data and source wavelet can be restructuredin some way to process pressure and vertical particle velocitydata. This may have implications, for instance, for attenuatingintrabed multiples by wave-theoretical methods.

In the following section, we will generalize the algorithms(4), (5), and (7) so that they are valid for any inhomogeneoussubsurface.

INTEGRAL RELATIONSHIP BETWEEN PHYSICAL ANDHYPOTHETICAL (DESIRED) EXPERIMENTS

I here establish an integral relationship between therecorded data excited by a marine source array in the physicalseismic experiment, containing all free-surface related mul-tiples, and the desired data with those multiples absent. Thedesired data are the data that would be recorded in a hypo-thetical seismic experiment from a monopole point sourcewith desired signature when the water layer extends upwardsto infinity. The geology below the water layer is naturally thesame in the physical and hypothetical seismic experiments.

Define a volume V enclosed by the surface S= S0+ SR withoutward-pointing normal vector n as depicted in Figure 1,where S0 is the air/water surface, and SR represents a hemi-sphere of radius R. Let the Cartesian coordinate be denoted byx= (χ, z), where χ= (x, y) represents horizontal coordinates.Consider now the marine seismic experiment with configura-tion as indicated in Figure 1a. The recorded pressure data atreceiver location xr from the marine source array at center lo-cation xs, where zs< zr , is denoted by p(xr , ω; xs). I assume asis usual that the source array can be treated as an equivalent ar-ray of N independent, noninteracting monopole sources, eachgenerating a spherical wave (see Amundsen, 2000; Ziolkowskiet al., 1982). The source term thus is

s(x, ω) = −N∑

i=1

ai (ω)ρ(x)

δ(x− xsi ), (11)

where ai is the effective (notional) source signature for thesource at position xsi . In addition, I assume that the verticalcomponent of the particle velocity vz(xr , ω; xs) is known, eitherfrom recording or by estimation. Below the water layer, thesolid medium (geology) may have any structural complexity.

The recorded pressure field obeys the wave equation

(∇ · ρ−1∇+ ω2K−1)p(x, ω) = s(x, ω), (12)

where K is compression modulus. The pressure field p vanishesat the free surface S0 at z= 0; that is,

p(χ, z= 0) = 0. (13)

The desired pressure wavefield that I propose to solve foris the response of the medium from a single monopole pointsource with desired signature a when the free surface is absent.I consider the hypothetical, desired state shown in Figure 1bfor seismic acquisition: an infinite water layer above the samegeology as in Figure 1a. The volumes V bounded by the surfaceSare identical in Figures 1a and 1b. However, note that S0 in thehypothetical experiment is an artificial, nonphysical boundary.

FIG. 1. Geometry of the physical and hypothetical seismic ex-periments. The surface S= S0+ SR with outward pointing nor-mal vector n encloses a volume V consisting of the water layerand the solid. (a) In the physical experiment, S0 is a free sur-face with vanishing pressure. The source is positioned at centerlocation xs, and the receiver is located at xr . The free surfaceis a perfect reflector for all upgoing waves, which are reflecteddownwards, giving rise to multiples. The proposed free-surfacedemultiple theory assumes that the receiver is located belowthe source depth level in the physical experiment. (b) In thehypothetical experiment, S0 is a nonphysical boundary: all up-going waves from the subsurface continue to propagate in theupward direction. No multiples are generated. The source is amonopole point source located at xr , and the receiver is locatedat xs.

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Elimination of Free-Surface Multiples 331

The desired pressure response obeys the wave equation

(∇ · ρ−1∇+ ω2K−1) p(x, ω) = s(x, ω) (14)

with source term

s(x, ω) = − a(ω)ρ(x)

δ(x− xr ), (15)

and is assumed to be recorded at location xs for the monopolepoint source located at xr . In the case when a= 1, p is theGreen’s function response of the subsurface in the hypothet-ical experiment. The desired pressure response obeys source-receiver reciprocity

p(xr , ω; xs) = p(xs, ω; xr ). (16)

In Appendix A, equation (A-9), I show that Rayleigh’s theo-rem applied in the volume V enclosed by the surface S0+ SR→∞as depicted in Figure 1 gives an integral equation describing therelationship between the physical and hypothetical states:

a(ω)p(xr , ω; xs)−N∑

i=1

ai (ω) p(xsi , ω; xr

)= −iωρ

∫S0

dS(χ) p(x, ω; xr )vz(x, ω; xs), (17)

where we have taken into account boundary condition(13). The surface SR does not contribute to the integralas we let R→∞. This is Sommerfeld’s radiation condition(Sommerfeld, 1954). Equation (17) was derived earlier byFokkema and van den Berg (1993) for use in their free-surfacedemultiple approach. Equation (17) is also the starting pointin this paper for derving the proposed free-surface demultiplemethod. Observe that equation (17) requires the vertical parti-cle velocity component vz at the free surface S0. It is thereforequite obvious that we should redatum vz to the receiver level tomake equation (17) in conformity with the recording geome-try. We then find that no information about the physical sourceand its effective source wavelets ai is required to estimate pfrom the physical data p and vz measured at the receiver level.The only source information we require is its center locationxs.

At this stage, note that we have some freedom where to closethe upper part of the volume V . Instead of closing V along thefree surface S0, we may close V along the receiver surface; thischoice will lead us to the same free-surface demultiple algo-rithm as we will find by closing along S0 (see Amundsen andIkelle, 2000). By closing the volume V along the receiver sur-face, it is readily recognized from Rayleigh’s reciprocity the-orem (given in Appendix A) that no information about thephysical source array is required: the source array is outside thevolume under consideration and therefore does not explicitlyenter the integral relationship between the fields in the physi-cal and hypothetical states (the source array effect is implicitlycontained in the boundary conditions). From this observation,we deduce that the free-surface demultiple theory is indepen-dent of source array characteristics. In this paper, however, Ihave chosen to expoit equation (17) to derive the demultiplealgoithm, as did Fokkema and van den Berg (1993). However,I attack equation (17) differently from Fokkema and van denBerg.

To proceed, there are two options. One option is to use theKirchhoff-Helmholtz integral relationship to redatum the ver-

tical component of the particle velocity. However, I here pro-ceed by performing redatuming in the wavenumber domainusing well-known relationships which are summarized in Ap-pendix B. As stated earlier, I assume that the source locationis above the receiver station depth level so that 0< zsi < zr fori = 1, . . . , N. I also assume (as discussed in the Introduction)that the pressure field is a sum of upgoing and downgoing wavesaccording to

p(xr , ω; xs) = u(xr , ω; xs)+ d(xr , ω; xs). (18)

Let κ= (kx, ky) be the wavenumbers conjugate to the horizon-tal coordinate χ= (x, y). Using Parceval’s identity∫ ∞−∞

dχ f (χ)h(χ) = 1(2π)2

∫ ∞−∞

dκF(−κ)H(κ), (19)

observe that equation (17) can be written as

a(ω)p(xr , ω; xs)−N∑

i=1

ai (ω) p(xsi , ω; xr

) = −iωρ(2π)−2

×∫ ∞−∞

dκP(−κ, z= 0, ω; xr )Vz(κ, z= 0, ω; xs).

(20)

Recall that capital letters are used for field quantities in thewavenumber domain: P is the Fourier transform of p, and Vz

is the Fourier transform of vz. Substituting Vz at z= 0 with thedowngoing pressure wavefield D at the receiver level zr > zsi(i = 1, . . . , N) according to equation (B-19), we obtain

a(ω)p(xr , ω; xs)−N∑

i=1

ai (ω) p(xsi , ω; xr

)= −iωρ(2π)−2

∫ ∞−∞

dκP(−κ, z= 0, ω; xr )

×{

2kz

ωρexp(−ikzzr )D(κ, zr , ω; xs)

+N∑

i=1

ai (ω)iωρ

exp(−ikzzsi

)exp

(−iκ · χsi

)}, (21)

where kz=√

k2 − κ2 is the vertical wavenumber. The last termon the right side of equation (21) appears because of the dis-continuity of the downgoing wavefield across the source depthlevel. Since P is a purely upgoing field, we realize from equa-tion (B-16) that

P(−κ, zsi , ω; xr

) = P(−κ, z= 0, ω; xr ) exp(−ikzzsi

).

(22)Furthermore, in the last term in equation (21), we make theidenification

p(xsi , ω; xr

)= 1

(2π)2

∫ ∞−∞

dκ exp(−iκ · χsi

)P(−κ, zsi , ω; xr

).

(23)

Hence, the second term on the left side of equation (21) cancelsthe last term on its right side. Equation (21) now states that

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332 Amundsen

a(ω)p(xr , ω; xs)

= 12π2

∫ ∞−∞

dκ(−ikz)P(−κ, zr , ω; xr )D(κ, zr , ω; xs),

(24)

or

a(ω)p(xr , ω; xs)

= − iωρ

2π2

∫ ∞−∞

dκP(−κ, zr , ω; xr )D(Vz)(κ, zr , ω; xs),

(25)

where we have introduced D(Vz), which is the downgoing con-stituent of the vertical component of the particle velocity, com-puted from Vz and P according to equation (10). Using Parce-val’s identity (19), equation (25) yields

a(ω)p(xr , ω; xs)

= −2iωρ∫ ∞−∞

dχ p(χ, zr , ω; xr )d(vz)(χ, zr , ω; xs).

(26)

Equation (26) gives the sought-after integral relationship be-tween the desired pressure field p with the free surface absentwhen both sources and receivers are at depth level zr , and therecorded pressure field and downgoing component of the verti-cal particle velocity field at depth level zr from a source locatedat center location xs. Note that no information about the effec-tive source wavelets ai (ω) of the marine source array is requiredto solve for the desired pressure field p. Equation (26), whichis valid for any inhomogeneous subsurface, generalizes equa-tion (7), which is valid only for a horizontally layered medium.

The desired field p in the hypothetical experiment can besplit into a direct wave contribution pd propagating directlyfrom the source to the receiver, and the wavefield pr reflectedfrom the subsurface,

p = pd + pr ; P = Pd + Pr . (27)

The direct wave contribution, which propagates in a homoge-neous medium and is independent of source location, simplyfollows from equation (B-12) by setting ai = a and N= 1, thatis,

Pd(−κ, zr , ω; xr ) = − a

2ikzexp(iκ · χr ). (28)

Observing that

a(ω)d(xr , ω; xs)

= 12π2

∫ ∞−∞

dκ(−ikz)Pd(−κ, zr , ω; xr )D(κ, zr , ω; xs)

= a

(2π)2

∫ ∞−∞

dκ exp(iκ · χr )D(κ, zr , ω; xs), (29)

and using equation (18), equation (24) yields

a(ω)u(xr , ω; xs)

= 12π2

∫ ∞−∞

dκ(−ikz)Pr (−κ, zr , ω; xr )D(κ, zr , ω; xs).

(30)

Introducing D(Vz) and using Parceval’s identity (19), equa-tion (30) is written in space domain as

a(ω)u(xr , ω; xs)

= −2iωρ∫ ∞−∞

dχ pr (χ, zr , ω; xr ) d(vz)(χ, zr , ω; xs).

(31)

Equation (31), which generalizes equation (4), gives the inte-gral relationship between the reflected field pr in the hypothet-ical experiment with both sources and receivers at depth levelzr , and upgoing component of pressure and downgoing compo-nent of vertical particle velocity at depth level zr from a sourcelocated at center location xs. Note again that no informationabout the wavelets ai (ω) is required to solve for pr .

Equations (24) and (31) are Fredholm integral equations ofthe first kind for the desired field. The reader is referred toTricomi (1957) or Antia (1991) for a discussion of numericalsolution techniques (such as quadrature or expansion methods)for Fredholm integral equations of the first kind. Such integralequations, in general, are ill-conditioned and their accurate so-lution may be difficult to obtain. In quadrature schemes, the in-tegral is approximated by a quadrature formula, and the result-ing system of algebraic equations is solved. Here, observe thatone system of equations is obtained by fixing the receiver coor-dinate and varying the source coordinate. In expansion meth-ods, the solution is approximated by an expansion in terms ofsome convenient basis functions. The coefficients of expansionmay be determined by minimizing some error norm. Anotherpossible solution technique can be to transform the integralequations to the wavenumber domain (as done below), wherethey may be easier to handle. More work is required to solvethis issue.

Wavenumber domain solution

In this subsection, I discuss how equation (31) can be trans-formed to the wavenumber domain. By Fourier transformingequation (31) over source coordinates χs and receiver coordi-nates χr with conjugate wavenumbers κs and κr , respectively,and making use of Parceval’s identity, we find

a(ω)U(κr , zr , ω;κs, zs)

= 12π2

∫ ∞−∞

dκ(−ikz)Pr (− κ, zr , ω;κr , zr )

× D(κ, zr , ω;κs, zs). (32)

Equation (32) leads to a system of equations that can be solvedfor Pr by keeping the wavenumber κr conjugate to the re-ceiver coordinate fixed while varying the wavenumber κs con-jugate to the source coordinate. The coupling between positivewavenumbers in the downgoing field with negative wavenum-bers in the desired field (and vice versa) reflects the autocor-relation process between the two fields.

A similar, but not equal, wavenumber domain scheme foreliminating the effect of the free surface in marine seismic datawas proposed by Ziolkowski et al. (1998, 1999). Ziolkowski andcoworkers subtract the direct wave (including its sea-surfaceghost) from the pressure and vertical component of particlevelocity before decomposition into upgoing and downgoingconstituents. The direct wave is modeled, requiring that the

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Elimination of Free-Surface Multiples 333

source signature is known or can be estimated. The algorithmof Ziolkowski and coworkers can be derived quite simply fromequation (32) by considering the downgoing wave D= Pd + Dr

to be the sum of the direct wave (including source ghost) Pd

and the downgoing part of the field Dr reflected from the sub-surface. For simplicity, consider the marine source to consistof one single element with signature a at location xs. The di-rect wave field (including the source ghost) is given in equation(B-13) as

Pd(κ, zr ; xs) = − a

2ikz{exp[ikz|zr − zs|]

− exp[ikz(zr + zs)]} exp(−iκ · χs). (33)

By performing a Fourier transform over the horizontal sourcecoordinates χs, I obtain

Pd(κ, zr , ω;κs, zs)

= − a

2ikz{exp[ikz|zr − zs|]− exp[ikz(zr + zs)]}

× (2π)2δ(κ+ κs). (34)

Inserting expression (34) into equation (32) with a=a and us-ing the Dirac delta function property∫ ∞

−∞dκδ(κ− κ′) f (κ) = f (κ′), (35)

yields

U(κr , zr , ω;κs, zs)

= −2ikz,sR(κs, zr , ω;κr , zr )Pd(κs, zr , ω;χs = 0, zs)

+ 12π2

∫ ∞−∞

dκ(−ikz)R(−κ, zr , ω;κr , zr )

× Dr (κ, zr , ω;κs, zs), (36)

where the vertical wavenumber kz,s=√

k2 − κ2s depends on the

horizontal wavenumbers conjugate to the horizontal source co-ordinates. Note that the source wavelet a is required to calcu-late Pd. In equation (36), I have introduced the Green’s func-tion reflection response

R= a−1 Pr . (37)

It can be shown that algorithm (36), which was first derivedby Ziolkowski and coworkers, is valid for any relative source-receiver depth in the water column, while algorithm (32) aspreviously stated requires that zr > zs. Usually marine data andin particular ocean-bottom seismic data are acquired with re-ceiver depth below source depth. For cases of practical interest,equation (32) therefore can be used for free-surface pressuredemultipling.

I conclude that when the pressure field recorded at a depthlevel zr > zs can be decomposed into upgoing and downgoingcomponents, both equations (32) and (36) can be used to elim-inate the effect of the free surface. The two equations providedifferent methods and prerequisites, however, for removingthe multiples. While method (32) is independent of the sourcewavelet and any special treatment of the direct field Pd, method(36) is not. Further, implicit in equation (32) is a source signa-ture deconvolution process which is not present in method (36).

SPECIAL CASE: HORIZONTALLY LAYERED MEDIUM

In a horizontally layered medium, the response is laterallyshift invariant with respect to horizontal source location χs.Consider equation (30), where we may set χs= 0 and

Pr (−κ, zr , ω; xr )= Pr (−κ, zr , ω;χ= 0, zr ) exp(iκ · χr ).

(38)By Fourier transforming equation (30) with respect to χr andinterchanging integrals, we find

a(ω)U(κr , zr , ω;χ = 0, zs)

= 12π2

∫ ∞−∞

dκ(−ikz)Pr (−κ, zr , ω;χ = 0, zr )

× D(κ, zr , ω;χ = 0, zs)∫ ∞−∞

dχr exp[iχr · (κ− κr )].

(39)

The last integral is recognized as the Dirac delta function:∫ ∞−∞

dχr exp[iχr · (κ− κr )] = (2π)2δ(κ− κr ). (40)

Performing the integration over wavenumbers, using the Diracdelta function property (35), and renaming κr by κ, we obtain

a(ω)U(κ, zr , ω;χs = 0, zs)

= −2iωρ Pr (−κ, zr , ω;χs = 0, zr )

× D(Vz)(κ, zr , ω;χs = 0, zs). (41)

Using that the response is symmetric with respect to wavenum-bers, the reflected part of the desired field is obtained by spec-tral deconvolution:

Pr (κ, zr , ω;χs = 0, zr )

= − a(ω)2iωρ

U(κ, zr , ω;χs = 0, zs)D(Vz)(κ, zr , ω;χs = 0, zs)

. (42)

Equation (42) is consistent with equation (5). Most often, onewants Pr at zr for a source at true depth level zs. This desiredfield is simply found by applying a phase shift correcting fordifference in source and receiver levels according to

Pr (κ, zr , ω;χs = 0, zs)

= − a(ω)2iωρ

U(κ, zr , ω;χs = 0, zs)D(Vz)(κ, zr , ω;χs = 0, zs)

exp[ikz(zr − zs)],

(43)

or

Pr (κ, zr , ω;χs = 0, zs)

= − a(ω)2iωρ

U(κ, zs, ω;χs = 0, zs)D(Vz)(κ, zr , ω;χs = 0, zs)

. (44)

Likewise, we find that the total desired field is

P(κ, zr , ω;χs = 0, zr )

= − a(ω)2iωρ

P(κ, zr , ω;χs = 0, zs)D(Vz)(κ, zr , ω;χs = 0, zs)

. (45)

Equation (45) is identical to equation (7). Possible differencein source and receiver levels are adjusted as

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334 Amundsen

P(κ, zr , ω;χs = 0, zs)

= − a(ω)2iωρ

P(κ, zr , ω;χs = 0, zs)D(Vz)(κ, zr , ω;χs = 0, zs)

exp[ikz(zr − zs)].

(46)

Observe that − 2iωρD(vz) exp[ikz(zr − zs)] is a multidimen-sional spiking deconvolution operator. This demultiple schememay be implemented as τ -p or frequency-wavenumber domainalgorithms. In the τ -p domain, a joint designature and multi-ple attenuation process is performed for each p-trace. In thefrequency-wavenumber domain, the process is performed foreach combination of frequency and wavenumber. Sønnelandet al. (1986) proposed a scheme similar to equations (44) forpressure designature and demultiple from data recorded fromvertical receiver arrays. However, their scheme was never fullyimplemented.

ADJUSTMENT FOR DIFFERENCE IN SOURCEAND RECEIVER DEPTH LEVELS

By solving the integral equations (31) or (32), we find the re-flection response of the earth for the hypothetical experimentwhen the source and receivers are at the same depth levels. Inthe physical experiment, the source is at a depth level zs abovethe receiver depth level at zr . In particular for ocean-bottomseismic data, the difference between zs and zr may be large, andone would therefore want to output the desired reflection re-sponse for a source at zs to meet with the true source-receiveracquisition geometry. This adjustment will make it easier tocompare reflection events in the time-space domain betweenthe recorded pressure data and the desired reflected part ofthe pressure data. By phase shifting the upgoing part of therecorded pressure field after wavefield decomposition in thewavenumber domain by the depth level difference zr − zs ac-cording to

U(κ, zs, ω; xs) = U(κ, zr , ω; xs) exp[ikz(zr − zs)], (47)

the inverse Fourier transformed field of U in equation (47), thatis u(χr , zs, ω; xs), can be used as input field to the free-surfacedemultiple process instead of u(χr , zr , ω; xs). The phase shift-ing process (47) of the upgoing field implies that the integralequations (31) and (32) for the desired field are adjusted to thedifference in source and receiver depth levels. Equation (31)now becomes

a(ω)u(χr , zs, ω; xs)

= −2iωρ∫ ∞−∞

dχ pr (χ, zs, ω; xr ) d(vz)(χ, zr , ω; xs),

(48)

whereas equation (32) yields

a(ω)U(κr , zs, ω;κs, zs)

= − 12π2iωρ

∫ ∞−∞

dκPr (−κ, zs, ω;κr , zr )

× D(Vz)(κ, zr , ω;κs, zs). (49)

FREE-SURFACE DEMULTIPLING OF THE VERTICALPARTICLE VELOCITY COMPONENT

Free-surface demultipling of the vertical particle velocitycomponent corresponds to finding the vertical particle veloc-ity field in the hypothetical experiment with the free surfaceabsent. By using the equation of motion (A-1), equation (26)yields the free-surface multiple elimination and source signa-ture deconvolution of the vertical particle velocity component:

a(ω)vz(xr , ω; xs)

= −2iωρ∫ ∞−∞

dχvz(χ, zr , ω; xr ) d(vz)(χ, zr , ω; xs).

(50)

NUMERICAL EXAMPLE FOR A HORIZONTALLYLAYERED MODEL

To illustrate and test the performance of the free-surfacedemultiple algorithm, I use line-source synthetic seismograms(2-D wave propagation) modeled by a frequency-wavenumberdomain algorithm over a horizontally layered medium. Themodeling algorithm allows the computation of the pressurefield and the vertical component of the particle velocity inthe water layer. These are the two datasets that are input tothe free-surface demultiple scheme. In addition, I model up-going and downgoing waves in the water layer, as well as thetotal and reflected part of the pressure field for an infinite waterhalfspace. These modeled components are used to verify thatall constituents of the free-surface demultiple algorithm worksfor a horizontally layered medium.

The model used in the numerical example is given in Ta-ble 1. The water layer thickness is 300 m. The source is locatedat depth level zs= 5 m below the free surface, and generatesa wavelet with a dominant frequency of approximately 15 Hz.The wavelet and its amplitude spectrum are shown in Figure 2.The receiver depth level is zr = 20 m. The minimum and maxi-mum offsets are ±3.2 km. The receiver spacing is 6.25 m. Therecording time is 2.2 s. The modeled pressure and vertical com-ponent of particle velocity are shown in Figure 3. For displaypurposes, only the offset range 0–1.6 km is shown.

Table 1. The layered model used to generate data. An elasticmedium is bounded above by a water layer and below by anelastic half-space.

Depth P-velocity S-velocity DensityLayer (m) (m/s) (m/s) (kg/m3)

1 0–300 1480 0 10302 300–305 1550 75 18003 305–340 1600 250 18504 340–405 1700 325 18705 405–430 1900 420 19506 430–500 1700 370 18307 500–525 2000 500 21008 525–960 1800 450 19809 960–1173 2000 700 200010 1173–1186 2538 1250 210011 1186–1458 2252 1200 216012 1458–1547 2309 1414 195013 1547–1567 2400 1570 215014 1567– 2600 1570 2250

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Elimination of Free-Surface Multiples 335

The first step in the demultiple procedure is to decomposethe pressure and particle velocity into upgoing and downgo-ing components. The decomposition is performed according toequations (8) and (10) when replaced by their 2-D equivalents,and requires that the velocity and density of water are known.The upgoing component of pressure and downgoing compo-nent of the vertical particle velocity field are shown in Figure 4.They are numerically in agreement with modeled upgoing anddowngoing waves (not shown).

The second step in the free-surface demultiple algorithm is tocompute the reflected part of the desired reflection response pr

from the upgoing and downgoing waves and a desired waveletaccording to the 2-D equivalent of equation (43). In this nu-merical exercise, the desired wavelet a is chosen equal to thewavelet a shown in Figure 2 to model the data in Figure 3. Thereflected part of the desired pressure field without any free-surface effect is shown in Figure 5a. No numerical instabilitieswere encountered in the data processing. To verify the perfor-

FIG. 2. Source time signature (top) and its amplitude spectrum(bottom). The source signature in this figure is the waveletused for modeling the data in Figure 3, as well as the desiredwavelet used for calculating without any free-surface effectthe reflected part of the pressure field in Figure 5a and the fullpressure field in Figure 6a.

mance of the demultiple and designature method, I have alsomodeled the reflected part of the pressure field. The modeledfield is shown in Figure 5b, confirming that the scheme for ahorizontally layered medium works satisfactorily.

Equation (46) shows that the total pressure field can be usedas input to the free-surface demultiple process instead of theupgoing part of the pressure field as suggested by equation (43).The difference between the two schemes is that the direct wavePd will be contained in the demultipled and designatured datawhen using equation (46). Figure 6a displays the demultiple,designature result according to equation (46). Observe that

FIG. 3. (a) Pressure field. (b) Vertical component of particle ve-locity. The data are modeled at receiver depth zr = 20 m belowa free surface. The source is at zs= 5 m.

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336 Amundsen

the direct wave (without any ghost) is recovered. The reflectedpart of the demultipled data is in agreement with the data inFigure 5a. To verify the numerical result, I have once moremodeled the pressure data including the direct wave whenthe free surface is absent. The modeled data are displayed inFigure 5b.

The elimination of the free surface from pressure recordingsrenders the data far more interpretable. Primary reflectionsare easy to identify in Figure 5a, whereas they are masked by

FIG. 4. (a) Upgoing component of pressure field displayed inFigure 3a, and (b) downgoing component of the vertical parti-cle velocity field displayed in Figure 3b. The time derivative ofthe downgoing component is the spiking deconvolution opera-tor which designatures the data and eliminates all free-surfacerelated multiples from the data.

multiples in Figure 3a. For the model I have used, primariesare extremely difficult to distinguish from multiples when thefree surface effect is present.

Note that the “wavelet” seen in the pressure data in Figure 3aand Figures 5a and 6a has different character. While the waveletin the “physical” pressure data in Figure 3a contains the ef-fect of source and receiver ghosts due to the free surface,the wavelet in the desired pressure in Figures 5a and 6a doesnot.

FIG. 5. (a) The reflected part of the pressure field without anyfree-surface effect, calculated from the data in Figure 4 accord-ing to equation (43). (b) Corresponding modeled reflected partof the pressure field for a water-layer half-space.

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Elimination of Free-Surface Multiples 337

FIG. 6. (a) The total pressure field (including the direct wave)without any free-surface effect, calculated from the data in Fig-ures 3a and 4b according to equation (46). (b) Correspondingmodeled pressure field for a water-layer half-space.

CONCLUSIONS

I have developed a new method for eliminating the effectof the free surface from marine seismic data measured in thewater column at a depth level below the depth level of thesource array. Recorded data are transformed during processingto new, desired data that would be recorded in a hypotheticalexperiment with no sea surface present. Source designature isan implicit part of the free-surface demultiple process. Ampli-tudes of primary reflections are preserved, and no knowledgeof the subsurface is required. The method does not need anyinformation, except location, about the physical source and

its signature, but requires that the vertical component of theparticle velocity is input to the free-surface demultiple pro-cess along with pressure recordings. These additional data areroutinely recorded in ocean-bottom seismic surveys. Hence,the proposed method is ideally designed for free-surface de-multiple and designature of ocean-bottom dual-sensor seismicdata. For free-surface demultiple and designature of conven-tionally towed streamer data, the best solution to the demul-tiple problem is to develop technology to record the verti-cal derivative of the pressure together with the pressure field.Since the direct wave and its source ghost is part of the free-surface demultiple, designature process, it is critical that thedirect arrival is properly measured for the method to worksuccessfully.

The data processing is divided into two main steps. First,the pressure recording and the vertical component of the par-ticle velocity are decomposed into upgoing and downgoingcomponents. The decomposition is most easily performed inthe frequency-wavenumber domain or the τ -p domain, withknown density and water velocity of sound. Second, an im-plicit integral equation inputing the pressure field or its upgoingcomponent, the downgoing component of the vertical particlevelocity field, and the desired wavelet related to a monopolepoint source must be solved to find the desired pressure re-sponse without the free-surface effect. By choosing the de-sired wavelet to be unity for all frequencies, the free-surfacedemultiple and designature process gives the Green’s functionresponse of the subsurface when the free-surface is absent.The desired wavelet is a user-defined wavelet that can be cho-sen by the geophysicist to give the processed data a waveletwith any desired phase and amplitude spectrum within the datafrequency-band.

I have shown that the free-surface demultiple and desig-nature scheme for the special case of a horizontally layeredmedium works satisfactorily.

ACKNOWLEDGMENTS

I thank Den norske stats oljeselskap a.s for allowing me topublish this paper. I also thank Anton Ziolkowski for the dis-cussions that were the trigger to this work.

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Elimination of Free-Surface Multiples 339

APPENDIX A

RAYLEIGH’S RECIPROCITY THEOREM

In this appendix, I first review Rayleigh’s reciprocity the-orem, which gives the relationship between two independentacoustic wavefields. I then apply the theorem to the case of in-terest in this paper: wavefields recorded in the physical seismicexperiment with the free-surface present and wavefields thatwould have been recorded in the hypothetical seismic experi-ment with the free-surface abscent.

General theory

The material presented in this subsection is well known, butI include it for completeness. Excellent books for readers seek-ing more details are Aki and Richards (1980), Wapenaar andBerkhout (1989), and Fokkema and van den Berg (1993). I in-troduce the basic acoustic equations which describe wave mo-tion in an inhomogeneous medium. The material parameters,wavefields, and sources are defined by: K (x)= compressionmodulus, ρ(x)= volume density of mass, v(x, ω)= particle ve-locity, p(x, ω)= acoustic pressure, f(x, ω)= volume densityof external force, and iv(x, ω)= volume density of volumeinjection.

A source of particular interest is a monopole point source,represented by iv =− (ρω2)−1aδ(x), where a is the source sig-nature and δ(x) is the Dirac delta function. The equation ofmotion states

∇p = iωρv+ f, (A-1)

and the constitutive relation is

iωK−1 p =∇ · v+ iωiv. (A-2)

The acoustic pressure field satisfies the wave equation

(∇ · ρ−1∇+ ω2K−1)p = s (A-3)

with source distribution

s= ω2iv +∇ · ρ−1f. (A-4)

Consider a volume V enclosed by the surface S with out-ward pointing normal vector n. In this volume, define twonon-identical acoustic wavefields denoted by the wavefieldsfor “state A” and “state B,” respectively. For the moment, donot specify the boundary conditions for the wavefields. By in-troducing the vector [pA(ρB)−1∇pB− pB(ρA)−1∇pA] and ap-plying Gauss’ theorem to its divergence∇ · [pA(ρB)−1∇pB−pB(ρA)−1∇pA], one readily obtains Rayleigh’s reciprocitytheorem

iω∮

SdS[pAvB

n − pBvAn

]=∫

VdV[pAsB − pBsA + ω2(1/K A − 1/K B)pA pB

− (1/ρA − 1/ρB)(∇pA) · (∇pB)], (A-5)

where vn denotes the normal component of the particle veloc-ity. We remark that the four last terms on the right side vanishwhen the media in the two states are identical in V (strictlyinside S), that is, K A= K B and ρA= ρB. For this situation, the

relationship between the two states is governed by differencesin source distributions and differences in boundary conditions.

Reciprocity between physical and hypothetical states

We now consider the physical and hypothetical experimentsas described in the main text and depicted in Figure 1 with vol-ume V and enclosing surface S= S0+ SR with outward pointingnormal vector.

In both states, SR is a hemisphere of radius R. Referring to theabove discussion, identify state A with the physical state (seeFigure 1a) with medium parameters, wave fields, and sources:

K A(x) = K (x),

ρA(x) = ρ(x),

vA(x) = v(x; xs),

pA(x) = p(x; xs),

sA(x) = −N∑

i=1

aiρ−1(x)δ

(x− xsi

)where xs is the center location of the marine source array. In thephysical experiment, S0 is the air/water surface with vanishingpressure

p(χ, z= 0) = 0. (A-6)

Furthermore, identify state B with the hypothetical state (seeFigure 1b) with medium parameters, wave fields, and sources:

K B(x) = K (x),

ρB(x) = ρ(x),

vB(x) = v(x; xr ),

pB(x) = p(x; xr ),

sB(x) = −aρ−1(x)δ(x− xr ).

In the hypothetical experiment, S0 is an artificial, nonphysical(nonreflecting) boundary with nonvanishing pressure

p(χ, z= 0) 6= 0. (A-7)

Inserting the above expressions into Rayleigh’s theorem (A-5)yields

iω∮

SdS(χ) [p(x; xs)vn(x; xr )− p(x; xr )vn(x; xs)]

= − a

ρ(xr )p(xr ; xs)+

N∑i=1

ai

ρ(xsi

) p(xsi ; xr

).

(A-8)

Letting the radius R go to infinity, the surface SR→∞ gives zerocontribution to the surface integral. This is Sommerfeld’s ra-diation condition (Sommerfeld, 1954). Furthermore, use theboundary conditions (A-6) and (A-7) on S0, where vn=− vz

is set since the z-axis is positive downwards. Thus, Rayleigh’s

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340 Amundsen

theorem gives the following integral relationship between thewavefields in the physical and hypothetical states:

a p(xr ; xs)−N∑

i=1

ai p(xsi ; xr )

= −iωρ∫

S0

dS(χ) p(x; xr )vz(x; xs), (A-9)

where ρ is density of water. Equation (A-9) is my starting pointfor deriving the free-surface demultiple scheme in the main text[see equation (17)].

Reciprocity in the hypothetical state

By letting both state A and state B have medium param-eters of the hypothetical state, that is, K A= K B and ρA= ρB,and interchaning source and receiver positions, that is, sA=−aρ−1δ(x− xs) and sB=−aρ−1δ(x− xr ), Rayleigh’s reci-procity theorem (A-5) simplifies to the well-known relation

p(xr ; xs) = p(xs; xr ), (A-10)

stating that the pressure at xr due to a monopole point sourcewith signature a at xs is identical to the pressure at xs due to amonopole point source with the same signature a at xr .

APPENDIX B

BASIC FORMULAS IN WAVENUMBER DOMAIN

In this paper, the temporal Fourier transform of a functionf (t) is defined as

F(ω) =∫ ∞−∞

dt exp(iωt) f (t), (B-1)

with inverse

f (t) = 12π

∫ ∞−∞

dω exp(−iωt)F(ω). (B-2)

With χ= (x, y) and κ= (kx, ky), define the 2-D spatial Fouriertransform by

F(κ) =∫ ∞−∞

dχ exp(−iκ · χ) f (χ), (B-3)

with inverse

f (χ) = 1(2π)2

∫ ∞−∞

dκ exp(iκ · χ)F(κ). (B-4)

In the frequency-wavenumber domain, omit the explicit de-pendence of different quantities on frequency and wavenum-bers. For instance, the pressure field recorded in the waterlayer p(χ, z, ω; xs) over an elastic layered medium from a pointsource at location xs is in the wavenumber domain denoted byP or P(z) with the understanding P= P(z)= P(κ, z, ω; xs).

The pressure P and particle velocity Vz are composed ofupgoing U and downgoing D waves according to (see, e.g.,Amundsen, 1993)

P = U + D, (B-5)

Vz = kz

ρω(D −U ) . (B-6)

The upgoing and downgoing wave consituents can be com-puted from P and Vz as

U = 12

(P − ρω

kzVz

), (B-7)

D = 12

(P + ρω

kzVz

). (B-8)

Observe from equation (B-5) that U and D are defined suchthat their sum gives the pressure field. The scaling of upgoingand downgoing waves, however, is not unique. Therefore in-troduce upgoing U (Vz) and downgoing D(Vz) waves whose sumgives Vz, that is,

Vz = U (Vz) + D(Vz), (B-9)

where

U (Vz) = 12

(Vz− kz

ρωP

), (B-10)

D(Vz) = 12

(Vz+ kz

ρωP

)(B-11)

The downgoing constituent D(Vz) of the vertical particle ve-locity component Vz is computed from the particle velocity it-self along with the pressure field according to equation (B-11).D(Vz) enters the integral relationship between the physical ma-rine experiment and the hypothetical marine experiment asa known field quantity along with the pressure recording [orits upgoing component which is computed according to equa-tion (B-7)].

For a homogeneous acoustic medium, the solution of theHelmholtz equation (12) with source term (11) is

P(z) = −N∑

i=1

ai

2ikzexp

(ikz

∣∣z− zsi

∣∣) exp(−iκ · χsi

).

(B-12)The solution (B-12) is the direct wave from each source tothe receiver when the free surface is absent. When the ho-mogeneous medium is bounded by a free surface at z= 0, thesolution of the Helmholtz equation is

P(z) ≡ Pd(z) = −N∑

i=1

ai

2ikz

{exp

[ikz

∣∣z− zsi

∣∣]− exp

[ikz(z+ zsi

)]}exp

(−iκ · χsi

). (B-13)

The solution (B-13) in the main body of the paper is defined asthe direct (incident) wavefield when the free surface is present.Observe that Pd simply is the sum of the waves traveling di-rectly from each source to the receiver and their sea-surfaceghosts.

In a homogeneous, source-free acoustic medium, upgoingand downgoing waves satisfy the wave equations

∂zU(z) = −ikzU(z), (B-14)

∂zD(z) = ikzD(z), (B-15)

with solutions

U(z) = exp[−ikz(z− z0)]U(z0), (B-16)

D(z) = exp[ikz(z− z0)]D(z0), (B-17)

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Elimination of Free-Surface Multiples 341

describing wavefield extrapolation of upgoing and downgoingwaves from depth level z0 to depth level z> z0.

When an array of N point sources are located at xsi(i = 1, . . . , N) in the water layer, the downgoing wavefield atdepth level z0 < zsi is related to the downgoing wavefield atdepth level z> zsi as

D(z0) = exp[−ikz(z− z0)]D(z)

+N∑

i=1

ai

2ikzexp

[−ikz(zsi − z0

)]exp

(−iκ · χsi

).

(B-18)

The last term appears because of the field discontinuity acrossthe source levels. Just below the free surface at z= 0 with van-ishing pressure, so that U =−D, the vertical particle velocityat depth level z= 0 is related to the downgoing wavefield atdepth level z> zsi (i = 1, . . . , N) according to

Vz(z= 0) = 2kz

ωρexp(−ikzz)D(z)

+N∑

i=1

ai

iωρexp

(−ikzzsi

)exp

(−iκ · χsi

).

(B-19)

Triangle relationships

In general, a “triangle relationship” defines a theoretical re-lation between three quantities in the water layer. When anyof two quantities are known, the third can be predicted fromtheory. A triangle relationship defines a theoretical relation be-tween the pressure, the vertical component of particle velocity,

and the source wavelets in the case when the receiver depthlevel is below the source depth level, zr > zsi (i = 1, . . . , N).This relationship has been used by Weglein and Secrest (1990)and Osen et al. (1995) to estimate the source wavelet fromrecordings of pressure and particle velocity, and by Amundsenet al. (1995) to estimate the particle velocity from the pres-sure when the source wavelets are known. In this appendix,I give the triangle relationship in the frequency-wavenumberdomain, where it reads (see, e.g., Amundsen, 1993)

N∑i=1

exp[ikz(zr − zsi

)]G−(zsi

)ai (ω)

= ikzG+(zr )P(zr )+ iωρG−(zr )Vz(zr )(zr > zsi

),

(B-20)

where

G±(z) = 1± exp(2ikzz). (B-21)

Here, G+(zr ) and G−(zr ) are receiver ghost operators thatwould be experienced by geophones and hydrophones, respec-tively, and G−(zsi ) is a source ghost operator.

Alternatively, the triangle relationship (B-20) may be givenin terms of upgoing and downgoing waves as

N∑i=1

exp[ikz(zr − zsi

)]G−(zsi

)ai (ω)

= 2ikz[D(zr )+ exp(2ikzzr )U(zr )](zr > zsi

).

(B-22)

Finally, note that equation (B-19) defines a triangle relation-ship between Vz(z= 0), D(zr ) and the source wavelets ai .

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