Deyan Draganov, Kees Wapenaar, Jan ThorbeckeDepartment of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628RX Delft, The Netherlands

With seismic interferometry we can simulate seismic shot records at point A as if from asource at point B by crosscorrelating the seismic responses recorded at A and B. Theseresponses can represent diffuse wavefields due to multiple scattering in deterministic ordiffuse media (Lobkis and Weaver, 2001; Derode et al., 2003) or due to uncorrelatednoise sources (Wapenaar et al., 2002; Shapiro and Campillo, 2004). Or the responsescan be due to transient sources in deterministic media (Schuster, 2001; Wapenaar et. al,2004; Bakulin and Calvert, 2006).We can also look at the recorded responses at A and B at the surface as due to activesources at the surface or due to passive sources in the subsurface. The latter was firstproposed by Claerbout (1968) where he showed that for 1-D acoustic media thereflection response can be simulated by the autocorrelation of the transmission response.Later Wapenaar et al. (2002, 2004) proved this for 3-D inhomogeneous lossless media(acoustic as well as elastic) using a one-way wavefield reciprocity theorem of thecorrelation type.Here, we show how to reconstruct the reflection response at the surface (the Green’sfunction in general) of a 3-D inhomogeneous medium (acoustic as well as elastic) fromthe crosscorrelation of transmission responses using relationships derived from a two-way wavefield reciprocity theorem (Wapenaar, 2004; van Manen et al., 2005; Wapenaarand Fokkema, 2006).

Introduction

Acoustic Seismic InterferometryFor an open acoustic inhomogeneous configuration with subdomain D limited by anarbitrary shaped boundary ∂D, which does not in general coincide with a physical one,Wapenaar and Fokkema (2006) show that the Green’s function G(xA,xB,ω) at point withcoordinate vector xA due to an impulsive source at a point with a coordinate vector xBcan be expressed in the frequency domain as

Fig. 2: (left) Time domain representation of the integrand in equation 2 for the model inFigure 1. Each trace represents the crosscorrelation result between the response at Bwith the response in A for each source position. (continues in next column)

Numerical simulations

Fig. 3: (left) Inhomogeneous subsurface model with uncorrelated noise sources in thesubsurface. The sources (stars) have random depth coordinate between 700 and 850 m.(right) The transmission response observed at the surface due to the simultaneouslyacting subsurface white-noise sources. Here we show only the first 3 s from a recordingwith a total length of 23 minutes.Relation 4 is applied to these data to reconstruct the reflection response at the freesurface at geophones every 10 m from 1200 till 6800 m.

Fig. 4: (left) Reconstructed reflection response at the free surface for the model inFigure 3 for variable point A and fixed point B. The simulated transient source at thesurface point B is at 4000 m.(right) Directly modeled reflection response for the model in Figure 3.

!

2" ˆ G xA ,xB ,#( ){ } = $1

j#% x( )ˆ G

*xA ,x,#( )&i

ˆ G xB ,x,#( )(&D'

$ &iˆ G xA ,x,#( ){ }

*ˆ G xB ,x,#( )

( ) * nid

2x, (1)

The approximation in relation 2 is limited mainly to the amplitudes. When the mediumoutside ∂D contains inhomogeneities, ghost events will appear in the reconstructedGreen’s function. These ghost events will be strongly weakened, though, whenintegration boundary ∂D is sufficiently irregular.In the case of transient uncorrelated noise sources along ∂D acting simultaneously, with

!

2" ˆ G xA,x

B,#( ){ } $ 2

%c

ˆ G *x

A,x,#( ) ˆ G x

B,x,#( )d

2x. (2)

&D'

where G(xA,x,ω) and ∂iG(xA,x,ω) are the recorded Green’s functions at the points A andB due to impulsive monopole and dipole sources, respectively, along the boundary ∂Dand ρ(x) the density in the medium along the boundary.If we assume the medium outside and along the boundary ∂D to be homogeneous withpropagation velocity c and density ρ, then equation 1 can be approximated by

Fig. 1: Homogeneous medium below a free surface at x3=0 (∂D0) with a singlediffractor at point C (x1,x3=0,600). There are observation recorders at points A(x1,x3=-500,100) and B (x1,x3=500,100), and multiple sources along the surface ∂D1,which is a semicircle with a radius of 800 m and centre at the origin. The propagationvelocity is c=2000 m/s. The solid lines indicate the Green’s function paths and thecircles indicate the stationary points at ∂D1.Relation 2 is applied to this model to reconstruct the Green’s function (in this case thereflection response) at point A due to an impulsive source at point B.

!

ˆ p obs

xA(B ),"( ) = ˆ G xA (B )

,x,"( ) ˆ N x,"( )d2x (3)

#D$

!

2" ˆ G xA ,xB ,#( ){ }S #( ) $2

%cˆ p

obsxA ,#( ){ }

*

ˆ p obsxB ,#( ) , (4)

the recorded responses at A and B from the simultaneously acting sources, we canrewrite equation 2 in the form

where <⋅> denotes spatial ensemble average and S(ω) - the power spectrum of the noise.

Fig. 2: (continued) (middle) Summed result over the sources along ∂D1. The maincontribution to the integral comes from traces around the stationary phase point (theFresnel zone). The contributions from the stationary points around φ=90º cancel eachother.(right) Zoomed in section of the comparison between the simulated and directlymodeled Green’s functions for the model in Figure 1.

Green’s Function Representations for Seismic Interferometry

Deyan Draganov, Kees Wapenaar, Jan ThorbeckeDepartment of Geotechnology, Delft University of Technology, Mijnbouwstraat 120, 2628RX Delft, The Netherlands

References

Bakulin, A., and R. Calvert, 2006, The virtual source method: theory and case study: Geophysics, accepted.Campillo, M., and A. Paul, 2003, Long-range correlations in the diffuse seismic coda: Science, 299, 547-549.Claerbout, J.F., 1968, Synthesis of a layered medium from its acoustic transmission response: Geophysics 33,264-269.Derode, A., E. Larose, M. Tanter, J. de Rosny, A. Tourin, M. Campillo, and M. Fink, 2003, Recovering theGreen’s function from field-field correlations in an open scattering medium: Journal of the Acoustical Society ofAmerica, 113, 2973-2976.Lobkis, O. I., and R. L. Weaver, 2001, On the emergence of the Green’s function in the correlations of a diffusefield: Journal of the Acoustical Society of America, 110, 3011-3017.Schuster, G.T., 2001, Theory of daylight/interferometric imaging: tutorial: 63rd Conference and Exhibition,EAGE, Extended abstracts A-32.Shapiro, N. M., and M. Campillo, 2004, Emergence of the broadband Rayleigh waves from correlations of theambient seismic noise: Geophysical Review Letters, 31, L07614-1 - L07614-4.van Manen, D.-J., J. O. A. Robertsson, and A. Curtis, 2005, Modeling of wave propagation in inhomogeneousmedia: Physical Review Letters, 94, 164301-1 - 164301-4.Wapenaar, C.P.A., J. W. Thorbecke, D. Draganov, and J. T. Fokkema, 2002, Theory of acoustic daylightimaging revisited: 72nd Annual International Meeting, SEG, Expanded abstracts ST 1.5.Wapenaar, C. P. A., 2004, Retrieving the elastodynamic Green’s function of an arbitrary inhomogeneousmedium by cross correlation: Physical Review Letters, 87, 134301-1 - 134301-4Wapenaar, C. P. A., D. Draganov, and J. W. Thorbecke, 2004, Relations between reflection and transmissionresponses of 3-D inhomogeneous media: Geophysical Journal International, 156, 179-194.Wapenaar, C. P. A., and J. T. Fokkema, 2006, Green’s functions representations for seismic interferometry:Geophysics, accepted.

We showed relationships that can be used to reconstruct the Green’s function at point Aas if from a source at point B from crosscorrelations of wavefield quantities observed atA and B due to sources on a a closed surface around the points. The relationships werederived using a two-way wavefield reciprocity theorem. In the elastic case, the observedquantities should be due to separate P- and S-wave sources along the integrationboundary.We showed with numerical modeling results that using the derived relations we cansuccessfully reconstruct the reflection response at the surface for acoustic media in thepresence of transient and noise sources as well as for elastic media in the presence oftransient sources.

Conclusions

Elastodynamic Seismic Interferometry

In the elastic case we also start with an inhomogeneous configuration with subdomain Dlimited by a arbitrary shaped boundary ∂D. Wapenaar and Fokkema (2006) show thatthe Green’s function Gυ,f

p,q(xA,xB,ω) at point with coordinate vector xA due to animpulsive source at a point with a coordinate vector xB can be expressed in thefrequency domain as

!

2" ˆ G p,q

# , fxA ,xB ,$( ){ } = % ˆ G p,i

# , fxA ,x,$( ){ }

*ˆ G q,ij

# ,hxB ,x,$( )

& ' (

)D*

+ ˆ G p,ij

# ,hxA ,x,$( ){ }

*ˆ G q,i

# , fxB ,x,$( )

+ , - n jd

2x, (5)

where the superscript φ stands for P-wave sources when K=0 and for S-wave sourceswith different polarizations when K=(1,2,3). I.e., at the observation points A and B werecord particle velocity due to separate P- and S-wave sources in the subsurface.If we assume the medium outside and along the boundary ∂D to be homogeneous withpropagation velocity for P-waves cp and for S-waves cs and density ρ, then equation 6can be approximated by

!

2" ˆ G p,q

# , fxA ,xB ,$( ){ } =

2

j$%&i

ˆ G p,K

# ,'xA ,x,$( ){ }

*ˆ G q,K

# ,'xB ,x,$( )

( ) *

+ , - nid

2x

&D. , (6)

where the superscript υ stands for the observed quantity (particle velocity), thesuperscripts f and h stand for the source quantity (force and deformation, respectively),the subscripts p and q indicate the receiver or source component. Note that Einstein’ssummation convention applies to repeated subscript indices. The Green’s functionsunder the integral are the recorded particle velocities at the points A and B due toimpulsive force and deformation sources, respectively, along the boundary ∂D.Equation 5 is difficult to apply in practice. That is why, Wapenaar (2004) and Wapenaarand Fokkema (2006) show further that the above relation can be rewritten as

!

2" ˆ G p,q

# , fxA ,xB ,$( ){ } %

2

&cK

ˆ G p,K

# ,'xA ,x,$( ){ }

*ˆ G q,K

# ,'xB ,x,$( )d2

x(D) , (7)

where cK stands for cp when K=0 and for cs when K=(1,2,3).The approximation in relation 7 concerns mainly the amplitudes, the phases not areaffected. Just like in the acoustic case, when the medium outside ∂D containsinhomogeneities, ghost events will appear in the reconstructed Green’s function. Theseghost events will be strongly weakened, though, when integration boundary ∂D issufficiently irregular.

Fig. 6: (left) This correlation result was produced by correlating the vertical particlevelocity (p=3) at variable xA at the free surface due to P-wave (K=0) subsurface sourceat variable x with the vertical particle velocity observed at fixed xB.(right) The same as for the left panel, but this time for a S-source.

Numerical simulations

Fig. 5: Inhomogeneous elasticsubsurface model with transient sourcesin the subsurface. The sources (stars)have random depth coordinate between700 and 800 m. At each source positionseparate P- and S-wave sources arefired. Relation 7 is applied to this modelto reconstruct the different componentsof the reflection response at the freesurface at geophones every 15 m from2100 till 5700 m.

Fig. 7: (left) Sum of the panels in Figure 6. This is the reconstructed vertical particlevelocity due to an impulsive vertical traction source at (x1,x3)=(0,3900).(right) Directly modeled vertical particle velocity response due to an impulsive verticaltraction source at (x1,x3)=(0,3900). Note that in this panel the direct and surface waveshave been removed.

Fig. 8: (left) Reconstructed vertical particle velocity due to an impulsive horizontaltraction source at (x1,x3)=(0,3900).(right) Directly modeled vertical particle velocity response due to an impulsivehorizontal traction source at (x1,x3)=(0,3900). Note that in this panel the direct andsurface waves have been removed.

AcknowledgmentsThis project is supported by the Netherlands Research Centre for Integrated Solid EarthScience (ISES), by the Technology Foundation STW, applied science division of NWOand the technology program of the Ministry of Economic Affairs (grant DTN4915). Wewould like to thank Prof. J. F. Zhang for providing the finite element code.

Green’s Function Representations for Seismic Interferometry(continued)