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

With seismic interferometry we can simulate seismic shot records at point A as if from a source at point B by crosscorrelating the seismic responses recorded at A and B. These responses can represent diffuse wavefields due to multiple scattering in deterministic or diffuse media (Lobkis and Weaver, 2001; Derode et al., 2003) or due to uncorrelated noise sources (Wapenaar et al., 2002; Shapiro and Campillo, 2004). Or the responses can 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 active sources at the surface or due to passive sources in the subsurface. The latter was first proposed by Claerbout (1968) where he showed that for 1-D acoustic media the reflection 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 the correlation type. Here, we show how to reconstruct the reflection response at the surface (the Green’s function in general) of a 3-D inhomogeneous medium (acoustic as well as elastic) from the crosscorrelation of transmission responses using relationships derived from a two-way wavefield reciprocity theorem (Wapenaar, 2004; van Manen et al., 2005; Wapenaar and Fokkema, 2006).

Introduction

Acoustic Seismic Interferometry

For an open acoustic inhomogeneous configuration with subdomain D limited by an arbitrary 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,) at point with coordinate vector xA due to an impulsive source at a point with a coordinate vector xB can be expressed in the frequency domain as

Fig. 2: (left) Time domain representation of the integrand in equation 2 for the model in Figure 1. Each trace represents the crosscorrelation result between the response at B with 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 the subsurface. The sources (stars) have random depth coordinate between 700 and 850 m. (right) The transmission response observed at the surface due to the simultaneously acting subsurface white-noise sources. Here we show only the first 3 s from a recording with a total length of 23 minutes.Relation 4 is applied to these data to reconstruct the reflection response at the free surface at geophones every 10 m from 1200 till 6800 m.

Fig. 4: (left) Reconstructed reflection response at the free surface for the model in Figure 3 for variable point A and fixed point B. The simulated transient source at the surface point B is at 4000 m.(right) Directly modeled reflection response for the model in Figure 3. In both pictures the pointer indicates the reflection event from the reflector below the sources.

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 medium outside D contains inhomogeneities, ghost events will appear in the reconstructed Green’s function. These ghost events will be strongly weakened, though, when integration boundary D is sufficiently irregular. In the case of transient uncorrelated noise sources along D acting simultaneously, with

2 ˆ G xA ,xB , 2

cˆ G * xA ,x, ˆ G xB ,x, d2x. (2)

D

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

Fig. 1: Homogeneous medium below a free surface at x3=0 (D0) with a single diffractor 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 propagation velocity is c=2000 m/s. The solid lines indicate the Green’s function paths and the circles indicate the stationary points at D1.

Relation 2 is applied to this model to reconstruct the Green’s function (in this case the reflection 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 obs xA , *

ˆ p obs xB , , (4)

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

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

Fig. 2: (continued) (middle) Summed result over the sources along D1. The main contribution to the integral comes from traces around the stationary phase point (the Fresnel zone). The contributions from the stationary points around =90º cancel each other.(right) Zoomed in section of the comparison between the simulated and directly modeled Green’s functions for the model in Figure 1.

Green’s function representations for seismic interferometry (continued)

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 the Green’s function from field-field correlations in an open scattering medium: Journal of the Acoustical Society of America, 113, 2973-2976.Lobkis, O. I., and R. L. Weaver, 2001, On the emergence of the Green’s function in the correlations of a diffuse field: 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 the ambient 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 inhomogeneous media: 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 daylight imaging 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 inhomogeneous medium 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 transmission responses 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 A as if from a source at point B from crosscorrelations of wavefield quantities observed at A and B due to sources on a a closed surface around the points. The relationships were derived using a two-way wavefield reciprocity theorem. In the elastic case, the observed quantities should be due to separate P- and S-wave sources along the integration boundary.We showed with numerical modeling results that using the derived relations we can successfully reconstruct the reflection response at the surface for acoustic media in the presence of transient and noise sources as well as for elastic media in the presence of transient sources.

Conclusions

Elastodynamic Seismic Interferometry

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

p,q(xA,xB,) at point with coordinate vector xA due to an impulsive source at a point with a coordinate vector xB can be expressed in the frequency domain as

2 ˆ G p,q , f xA ,xB , ˆ G p,i

, f xA ,x, * ˆ G q,ij ,h xB ,x,

D

ˆ G p,ij ,h xA ,x, * ˆ G q,i

, f xB ,x, n jd2x, (5)

where the superscript stands for P-wave sources when K=0 and for S-wave sources with different polarizations when K=(1,2,3). I.e., at the observation points A and B we record 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 with propagation velocity for P-waves cp and for S-waves cs and density , then equation 6 can be approximated by

2 ˆ G p,q , f xA ,xB ,

2

ji

ˆ G p,K , xA ,x, * ˆ G q,K

, xB ,x,

nid

2xD , (6)

where the superscript stands for the observed quantity (particle velocity), the superscripts 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’s summation convention applies to repeated subscript indices. The Green’s functions under the integral are the recorded particle velocities at the points A and B due to impulsive force and deformation sources, respectively, along the boundary D. Equation 5 is difficult to apply in practice. That is why, Wapenaar (2004) and Wapenaar and Fokkema (2006) show further that the above relation can be rewritten as

2 ˆ G p,q , f xA ,xB ,

2

cKˆ G p,K

, xA ,x, * ˆ G q,K , xB ,x, d2x

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 are affected. Just like in the acoustic case, when the medium outside D contains inhomogeneities, ghost events will appear in the reconstructed Green’s function. These ghost events will be strongly weakened, though, when integration boundary D is sufficiently irregular.

Fig. 6: (left) This correlation result was produced by correlating the vertical particle velocity (p=3) at variable xA at the free surface due to P-wave (K=0) subsurface source at 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 elastic subsurface model with uncorrelated noise sources in the subsurface. The sources (stars) have random depth coordinate between 700 and 800 m. At each source position separate P- and S-wave sources are fired. Relation 7 is applied to this model to reconstruct the different components of the reflection response at the free surface at geophones every 15 m from 2100 till 5700 m.

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

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

Acknowledgments

This project is supported by the Netherlands Research Centre for Integrated Solid Earth Science (ISES), by the Technology Foundation STW, applied science division of NWO and the technology program of the Ministry of Economic Affairs (grant DTN4915). We would like to thank Prof. J. F. Zhang for providing the finite element code.