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Greens function representation for seismic interferometry by deconvolution

Kees Wapenaar, Elmer Ruigrok, Joost van der Neut, Deyan Draganov, Jurg Hunziker, Evert Slob, Jan

Thorbecke, Delft University of Technology, and Roel Snieder, Colorado School of Mines

SUMMARY

Seismic interferometry by multi-dimensional deconvolution(MDD) has recently been proposed as an alternative to thecrosscorrelation method. Interferometry by MDD circumventsseveral of the underlying assumptions of the correlationmethod. We discuss a Greens function representation ofthe convolution type for seismic interferometry by MDD andillustrate it with a numerical example.

INTRODUCTION

Despite the strength of seismic interferometry to retrievenew seismic responses by crosscorrelating observations atdifferent receiver locations, the method relies on a number of

assumptions which are not always fulfilled in practice. Somepractical circumstances that may hamper interferometryby crosscorrelation are: one-sided illumination, irregularsource distribution, varying source spectra, multiples in theilluminating wave field, intrinsic losses, etc. To account forsome or more of these effects, several authors have proposedinterferometry by deconvolution, using a variety of approaches(Schuster and Zhou, 2006; Snieder et al., 2006; Mehta etal., 2007; Vasconcelos and Snieder, 2008; Wapenaar et al.,2008a, 2008b; van der Neut and Bakulin, 2009; Berkhout andVerschuur, 2009; van Groenestijn and Verschuur, 2010). Herewe discuss a Greens function representation of the convolutiontype as a unifying framework for seismic interferometry bymulti-dimensional deconvolution (MDD).

GREENS FUNCTI ON REPRESE NTATIONS

Consider an arbitrary inhomogeneous medium with acousticpropagation velocity c(x) and mass density (x) (where x =(x1, x2, x3) is the Cartesian coordinate vector). In this mediumwe consider an arbitrary closed surface S, with outward point-ing normal vector n= (n1, n2, n3), enclosing a volume V. Weconsider two points in V, denoted by coordinate vectors xAandxB , see Figure 1a. We define the Fourier transform of a time-dependent functionu(t) as u() =

exp(jt)u(t)dt, with

j the imaginary unit and the angular frequency. Assumingthe medium in V is lossless, the correlation-type representa-tion for the acoustic Greens function between xA and xB inV reads (Wapenaar et al., 2005; van Manen et al., 2005)

2{G(xB ,xA, )}=

S

1

j(x)

iG(xB ,x, )G

(xA, x, )

G(xB ,x, )iG(xA, x, )nidx (1)

(Einsteins summation convention applies to repeated lowercase Latin subscripts). This representation is the basic expres-sion for seismic interferometry (or Greens function retrieval)by crosscorrelation. The asterisk denotes complex conjuga-tion, hence, the products on the right-hand side correspondto crosscorrelations in the time domain of observations at tworeceivers at xA and xB . Representation 1 is exact, hence, itaccounts not only for the direct wave, but also for primary andmultiply scattered waves.

Ax

( , , )B A

G x x

( , , )B

G x x

( , , )A

G x x

a)

n

x

Bx

V

S

Sx

Bx

( , , )B S

G x x

( , , )

BG x x

x

b)

( , , )S

G x x

Sn

V

Fig. 1: (a) Configuration for the correlation-type Greens func-tion representation (equation 1). The medium in V is as-sumed lossless. The rays represent full responses, includingprimary and multiple scattering due to inhomogeneities insideas well as outside S. (b) Configuration for the convolution-type Greens function representation (equation 2). Here the

medium does not need to be lossless. The bar in G refers to areference state with possibly different boundary conditions atS and/or different medium parameters outside S.

Next, we consider a convolution-type representation for theGreens function. We slightly modify the configuration by tak-ing xA outside S and renaming it xS , see Figure 1b. For thisconfiguration the convolution-type representation is given by

G(xB, xS , ) =S

1j(x)

i G(xB ,x, )G(x,xS , )

G(xB , x, )iG(x, xS , )nidx. (2)

The bar in G is usually omitted because G and G are usuallydefined in the same medium throughout space. Here thebar is introduced to denote a reference state with possiblydifferent boundary conditions at S and/or different mediumparameters outside S (but in V the medium parameters forG are the same as those for G). Due to the absence of

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Greens function representation for interferometry by deconvolution

Bx

( )i

Sx

( )( , , )

i

B Su tx x

in ( )( , , )i

A Su tx x

d( , , )

B AG tx x

recS

a)

V

Ax

in ( )( , , )

i

A Su tx x

out ( )( , , )

i

B Su tx x

out

d ( , , )B AG tx x

Bx

( )i

Sx

recS

Ax

b)

V

Fig. 2: (a) Illustration of the convolutional model (equation12), underlying interferometry by MDD for the situation ofdirect-wave interferometry. (b) Idem, for reflected-wave in-terferometry.

RELATION WITH THE CORRELATION METHOD

To make the relation with the correlation method more trans-parent, we crosscorrelate both sides of equation 12 with

uin(xA, x

(i)S , t) (with x

A on Srec) and take the sum over all

sources, as proposed by van der Neut et al. (2010). This gives

C(xB , x

A, t) =

Srec

Gd(xB , xA, t)(xA,x

A, t) dxA, (13)

where

C(xB, x

A, t) (14)

=i

u(xB , x(i)S , t)uin(xA,x

(i)S ,t)

=i

G(xB , x(i)S , t)Gin(xA,x

(i)S ,t)S(i)(t),

(xA, x

A, t) (15)

=i

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

(i)S ,t)

=i

Gin(xA, x(i)S , t)Gin(xA,x

(i)S ,t)S(i)(t),

withS(i)(t) =s(i)(t)s(i)(t). (16)

x

recS

Ax '

in ( )( , , )i

A Su tx x

in ( )( , , )iA S

u tx x'

( )i

Sx

c)

V

Fig. 3: Fields involved in the illumination function(xA, x

A, t) (equation 15).

The correlation function C(xB, x

A, t) defined in equation 14

is the traditional interferometry relation (for example as em-ployed in the virtual source method of Bakulin and Calvert(2006)). Equation 13 shows that this correlation function is

proportional to the Greens function Gd(xB ,xA, t), smearedin space and time by (xA, x

A, t). According to equation 15,

(xA, x

A, t) is the crosscorrelation of the inward propagating

wave fields at xA and x

A, summed over the sources, see Fig-

ure 3. We call (xA,x

A, t) the illumination function (van der

Neut et al., 2010). If we would have a regular distribution ofsources with equal autocorrelation functions S(t) along a largesource boundary, and if the medium between the source andreceiver boundaries was homogeneous and lossless, the illumi-nation function would approach (xA, x

A, t) (c/2)2(xA

xA)S(t) [here (xA x

A) is defined for xA and x

A both onSrec; the approximation sign accounts for the fact that in real-ity the illumination function is spatially band-limited becausethe correlation in equation 15 does not compensate for evanes-cent waves]. Hence, for this situation, equation 13 would sim-plify to C(xB , x

A, t) (c/2)2Gd(xB , x

A, t) S(t), so the

Greens function Gd(

xB ,

x

A, t) could be simply obtained bydeconvolving the correlation function C(xB , x

A, t) forS(t). Inpractice, there are many factors that make the illuminationfunction (xA, x

A, t) deviate from a spatial delta function.Among these factors are the irregularity of the source distri-bution (Wapenaar et al., 2008b; van der Neut et al., 2010),medium inhomogeneities (van der Neut and Bakulin, 2009), afinite aperture (Schuster and Zhou, 2006), asymmetric illumi-nation of saltflanks (van der Neut and Thorbecke, 2009) or ofinter-well reflectors (Minato et al., 2009), multiple reflectionsin the illuminating wavefield (Wapenaar and Verschuur, 1996;Amundsen, 1999), intrinsic losses (Slob et al., 2007; Hunzikeret al., 2009; Fan et al., 2009), etc. In all those cases equation13 needs to be inverted by MDD, i.e., the effects of the illu-mination function (xA, x

A, t) need to be removed from the

correlation function C(xB , x

A, t) to obtain the Greens func-

tion Gd(xB , xA, t).

This is illustrated with a numerical example in Figure 4. TheNorth-South array in central USA in Figure 4a represents Srec,the East-West array a number of xB positions, and the bluedots along the East coast represent an irregular distribution

of source positions x(i)S

. Figure 4b is the illumination func-tion (xA, x

A, t) for fixed x

A (the red dot in Figure 4a) andvariable xA (the North-South array in Figure 4a). Figures 4cand d show in red the correlation function C(xB, x

A, t) and

the deconvolution result Gd(xB , x

A, t), respectively, for fixedxA (the red dot) and variable xB (the East-West array in Fig-

ure 4a). The obtained Greens function Gd(xB, x

A, t) matches

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Greens function representation for interferometry by deconvolution

the reference solution (in black) significantly better than thecorrelation function.

Note that the illumination function plays a similar role ininterferometry by MDD as the point-spread function (orspatial resolution function) in seismic migration (Schuster

and Hu, 2000; Gelius et al., 2002). The point-spread functionis defined as the migration result of the response of a singlepoint scatterer and as such is a useful tool to asses migrationresults in relation with geological parameters, backgroundmodel, acquisition parameters, etc. Moreover, it is sometimesused in migration deconvolution to improve the spatialresolution (Hu et al., 2001). An important difference is thatthe illumination function is obtained from measured responseswhereas the point-spread function is modeled in a backgroundmedium. As a result, the illumination function accountsmuch more accurately for the distorting effects of the mediuminhomogeneities, including multiple scattering.

NOISE SOURCES

We show that equation 13 also holds for the situation of simul-taneously acting uncorrelated noise sources. To this end, wedefine the correlation function and the illumination function,respectively, as

C(xB ,x

A, t) = u(xB , t)uin(xA,t), (17)

(xA,x

A, t) = uin(xA, t)u

in(xA,t), (18)

where the noise responses are defined as

uin(xA, t) =i

Gin(xA, x(i)S , t)N(i)(t), (19)

uin(xA, t) =j

Gin(xA, x(j)S , t)N(j)(t), (20)

u(xB , t) =j

G(xB , x(j)S , t)N(j)(t), (21)

and where the noise signals are mutually uncorrelated, accord-ing to

N(j)(t)N(i)(t)= ijS(i)(t). (22)

Upon substitution of equations 19 21 into equations 17and 18, using equation 22, it follows that the correlationfunction and the illumination function as defined in equations17 and 18 are identical to those defined in equations 14 and15. Hence, whether we consider transient or noise sources,equation 13 is the relation that needs to be inverted by MDDto resolve the Greens function Gd(xB, xA, t). It is beyondthe scope of this paper to discuss the numerical aspects of thisinversion.

CONCLUSIONS

We have derived a Greens function representation for seismic

interferometry by MDD. It appears that the correlation func-tion, used in correlation interferometry, can be written as thetrue Greens function, smeared in space and time by an il-lumination function. Similar as the correlation function, theillumination function is obtained directly from the measureddata. Interferometry by MDD removes the illumination func-tion from the correlation function and thus compensates for theeffects of one-sided illumination, irregular source distribution,intrinsic losses, etc. The theory discussed here covers most ofthe configurations usually considered, except that for passivereflected-wave interferometry some modifications are required,see van der Neut et al. (2010) for a further discussion.

Longitude [deg]

Latitude[deg]

Configuration

120oW 100

oW 80

oW 60

oW

20oN

30oN

40oN

50oN

54321ZYXWVUTSRQPONMLKJ

I

MCCM

WDCGMR

ISATUL1

MSOSDCOWRAK

WVOR

a)

Offset [km]

Time[s]

Illumination Function

600 400 200 0 200 400 600 800

200

150

100

50

0

50

100

150

b)

c)

d)

Fig. 4: Illustration of interferometry by MDD for surfacewaves. (a) Configuration. (b) Illumination function. (c)Crosscorrelation result (red) compared with reference solution(black). (d) MDD result.

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Greens function representation for interferometry by deconvolution

References

Amundsen, L., 1999, Elimination of free surface-related multiples without need of the source wavelet: 69th Annual InternationalSEG Meeting, Expanded Abstracts, 10641067.

Bakulin, A. and R. Calvert, 2006, The virtual source method: Theory and case study: Geophysics, 71, SI139SI150.

Berkhout, A. J. and D. J. Verschuur, 2009, Integrated imaging with active and passive seismic data: 79th Annual InternationalSEG Meeting, Expanded Abstracts, 15921596.

Fan, Y., R. Snieder, and J. Singer, 2009, 3-D controlled source electromagnetic (CSEM) interferometry by multi-dimensionaldeconvolution: 79th Annual International SEG Meeting, Expanded Abstracts, 779784.

Gelius, L.-J., I. Lecomte, and H. Tabti, 2002, Analysis of the resolution function in seismic prestack depth imaging: GeophysicalProspecting, 50, 505515.

Hu, J., G. T. Schuster, and P. Valasek, 2001, Poststack migration deconvolution: Geophysics, 66, 939952.

Hunziker, J., E. Slob, and K. Wapenaar, 2009, Controlled source electromagnetic interferometry by multidimensional deconvo-lution: spatial sampling aspects: 71rst Annual International EAGE Meeting, Expanded Abstracts, P074.

Mehta, K., R. Snieder, and V. Graizer, 2007, Extraction of near-surface properties for a lossy layered medium using thepropagator matrix: Geophysical Journal International, 169, 271280.

Minato, S., T. Matsuoka, T. Tsuji, D. Draganov, J. Hunziker, and K. Wapenaar, 2009, Application of seismic interferometry bymultidimensional deconvolution to crosswell seismic reflection using singular-value decomposition: 79th Annual InternationalSEG Meeting, Expanded Abstracts, 16881692.

Schuster, G. T. and J. Hu, 2000, Greens function for migration: Continuous recording geometry: Geophysics, 65, 167175.

Schuster, G. T. and M. Zhou, 2006, A theoretical overview of model-based and correlation-based redatuming methods: Geo-physics, 71, SI103SI110.

Slob, E. and K. Wapenaar, 2007, GPR without a source: Cross-correlation and cross-convolution methods: IEEE Transactionson Geoscience and Remote Sensing, 45, 25012510.

Slob, E., K. Wapenaar, and R. Snieder, 2007, Interferometry in dissipative media: Adressing the shallow sea problem for SeabedLogging applications: 77th Annual International SEG Meeting, Expanded Abstracts, 559563.

Snieder, R., J. Sheiman, and R. Calvert, 2006, Equivalence of the virtual-source method and wave-field deconvolution in seismicinterferometry: Physical Review E, 73, 06662010666209.

van der Neut, J. and A. Bakulin, 2009, Estimating and correcting the amplitude radiation pattern of a virtual source: Geophysics,74, SI27SI36.

van der Neut, J., E. Ruigrok, D. Draganov, and K. Wapenaar, 2010, Retrieving the earths reflection response by multi-dimensional deconvolution of ambient seismic noise: 72nd Annual International EAGE Meeting, Expanded Abstracts, P406.

van der Neut, J. and J. Thorbecke, 2009, Resolution function for controlled-source seismic interferometry: a data-drivendiagnosis: 79th Annual International SEG Meeting, Expanded Abstracts, 40904094.

van Groenestijn, G. J. A. and D. J. Verschuur, 2010, Estimation of primaries by sparse inversion from passive seismic data:Geophysics, 75, (in press).

van Manen, D.-J., J. O. A. Robertsson, and A. Curtis, 2005, Modeling of wave propagation in inhomogeneous media: PhysicalReview Letters, 94, 16430111643014.

Vasconcelos, I. and R. Snieder, 2008, Interferometry by deconvolution: Part 1 - Theory for acoustic waves and numericalexamples: Geophysics, 73, S115S128.

Wapenaar, C. P. A. and D. J. Verschuur, 1996, Processing of ocean bottom data: Presented at the The Delphi Acquisition

Project, Volume I, Delft University of Technology.Wapenaar, K., J. Fokkema, and R. Snieder, 2005, Retrieving the Greens function in an open system by cross-correlation: a

comparison of approaches (L): Journal of the Acoustical Society of America, 118, 27832786.

Wapenaar, K., E. Slob, and R. Snieder, 2008a, Seismic and electromagnetic controlled-source interferometry in dissipativemedia: Geophysical Prospecting, 56, 419434.

Wapenaar, K., J. van der Neut, and E. Ruigrok, 2008b, Passive seismic interferometry by multi-dimensional deconvolution:Geophysics, 73, A51A56.

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EDITED REFERENCES

Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2010

SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata foreach paper will achieve a high degree of linking to cited sources that appear on the Web.

REFERENCES

Amundsen, L., 1999, Elimination of free surface-related multiples without need of the source wavelet:69th Annual International SEG Meeting, Expanded Abstracts, 1064-1067.

Bakulin, A., and R. Calvert, 2006, The virtual source method: Theory and case study: Geophysics, 71, no.4, SI139SI150, doi:10.1190/1.2216190.

Berkhout, A. J., and D. J. Verschuur, 2009, Integrated imaging with active and passive seismic data: 79thAnnual International Meeting, SEG, Expanded Abstracts, 1592-1596.

Fan, Y., R. Snieder, and J. Singer, 2009, 3-D controlled source electromagnetic (CSEM) interferometryby multi-dimensional deconvolution: 79th Annual International Meeting, SEG, Expanded Abstracts,779-784.

Gelius, L.-J., I. Lecomte, and H. Tabti, 2002, Analysis of the resolution function in seismic prestack depthimaging: Geophysical Prospecting, 50, no. 5, 505515, doi:10.1046/j.1365-2478.2002.00331.x.

Hu, J., G. T. Schuster, and P. Valasek, 2001, Poststack migration deconvolution: Geophysics, 66, 939952, doi:10.1190/1.1444984.

Hunziker, J., E. Slob, and K. Wapenaar, 2009, Controlled source electromagnetic interferometry bymultidimensional deconvolution: spatial sampling aspects: 71st Annual International EAGEMeeting, Expanded Abstracts, P074.

Mehta, K., R. Snieder, and V. Graizer, 2007, Extraction of near-surface properties for a lossy layeredmedium using the propagator matrix: Geophysical Journal International, 169, no. 1, 271280,doi:10.1111/j.1365-246X.2006.03303.x.

Minato, S., T. Matsuoka, T. Tsuji, D. Draganov, J. Hunziker, and K. Wapenaar, 2009, Application ofseismic interferometry by multidimensional deconvolution to crosswell seismic reflection using

singular-value decomposit ion: 79th Annual International Meeting, SEG, Expanded Abstracts, 1688-1692.

Schuster, G. T., and J. Hu, 2000, Green's function for migration: Continuous recording geometry:Geophysics, 65, 167175, doi:10.1190/1.1444707.

Schuster, G. T., and M. Zhou, 2006, A theoretical overview of model-based and correlation-basedredatuming methods: Geophysics, 71, no. 4, SI103SI110, doi:10.1190/1.2208967.

Slob, E., and K. Wapenaar, 2007, GPR without a source: Cross-correlation and cross-convolutionmethods: IEEE Transactions on Geoscience and Remote Sensing, 45, no. 8, 25012510,doi:10.1109/TGRS.2007.900995.

Slob, E., K. Wapenaar, and R. Snieder, 2007, Interferometry in dissipative media: Adressing the shallow

sea problem for Seabed Logging applications: 77th Annual International Meeting, SEG, ExpandedAbstracts, 559-563.

Snieder, R., J. Sheiman, and R. Calvert, 2006, Equivalence of the virtual-source method and wave-fielddeconvolution in seismic interferometry: Physical Review E: Statistical, Nonlinear, and Soft MatterPhysics, 73, no. 6, 066620, doi:10.1103/PhysRevE.73.066620.PubMed

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Van der Neut, J., and A. Bakulin, 2009, Estimating and correcting the amplitude radiation pattern of avirtual source: Geophysics, 74, no. 2, SI27SI36, doi:10.1190/1.3073003.

Van der Neut, J., E. Ruigrok, D. Draganov, and K. Wapenaar, 2010, Retrieving the earth's reflectionresponse by multi-dimensional deconvolution of ambient seismic noise: 72nd Annual InternationalEAGE Meeting, Expanded Abstracts, P406.

Van der Neut, J., and J. Thorbecke, 2009, Resolution function for controlled-source seismicinterferometry: a data-driven diagnosis: 79th Annual International Meeting, SEG, ExpandedAbstracts, 4090-4094.

Van Groenestijn, G. J. A., and D. J. Verschuur, 2010, Estimation of primaries by sparse inversion frompassive seismic data: Geophysics, 75, in press.

van Manen, D.-J., J. O. A. Robertsson, and A. Curtis, 2005, Modeling of wave propagation ininhomogeneous media : Physical Review Letters, 94, no. 16, 164301,doi:10.1103/PhysRevLett.94.164301.PubMed

Vasconcelos, I., and R. Snieder, 2008, Interferometry by deconvolution: Part 1 - Theory for acousticwaves and numerical examples: Geophysics, 73, no. 3, S115S128, doi:10.1190/1.2904554.

Wapenaar, C. P. A., and D. J. Verschuur, 1996, Processing of ocean bottom data: The Delphi AcquisitionProject, Volume I, Delft University of Technology.

Wapenaar, K., J. Fokkema, and R. Snieder, 2005, Retrieving the Green's function in an open system bycross-correlation: a comparison of approaches: The Journal of the Acoustical Society of America,118, no. 5, 27832786, doi:10.1121/1.2046847.

Wapenaar, K., E. Slob, and R. Snieder, 2008a, Seismic and electromagnetic controlled-sourceinterferometry in dissipative media : Geophysical Prospecting, 56, no. 3, 419434,doi:10.1111/j.1365-2478.2007.00686.x.

Wapenaar, K., J. van der Neut, and E. Ruigrok, 2008b, Passive seismic interferometry by multi-dimensional deconvolution: Geophysics, 73, no. 6, A51A56, doi:10.1190/1.2976118.

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