The methodology of Green's function retrieval by cross-correlation has led to many interesting applications for passive and controlled-source acoustic measurements. In all applications, a virtual source is created at the position of a receiver. Here a method is discussed for Green's function retrieval from controlled-source reflection data, which circumvents the requirement of having an actual receiver at the position of the virtual source. The method requires, apart from the reflection data, an estimate of the direct arrival of the Green's function. A single-sided three-dimensional (3D) Marchenko equation underlies the method. This equation relates the reflection response, measured at one side of the medium, to the scattering coda of a so-called focusing function. By iteratively solving the 3D Marchenko equation, this scattering coda is retrieved from the reflection response. Once the scattering coda has been resolved, the Green's function (including all multiple scattering) can be constructed from the reflection response and the focusing function. The proposed methodology has interesting applications in acoustic imaging, properly accounting for internal multiple scattering.
REFERENCES
1.
G. L.
Lamb
, Elements of Soliton Theory
(John Wiley and Sons, Inc.
, New York
, 1980
), pp. 46
–67
.2.
R.
Burridge
, “The Gelfand-Levitan, the Marchenko, and the Gopinath-Sondhi integral equations of inverse scattering theory, regarded in the context of inverse impulse-response problems
,” Wave Motion
2
, 305
–323
(1980
).3.
D. B.
Ge
, “An iterative technique in one-dimensional profile inversion
,” Inverse Probl.
3
, 399
–406
(1987
).4.
K.
Chadan
and P. C.
Sabatier
, Inverse Problems in Quantum Scattering Theory
(Springer
, Berlin
, 1989
), Chaps. 5 and 7.5.
J. H.
Rose
, “ ‘Single-sided’ focusing of the time-dependent Schrödinger equation
,” Phys. Rev. A
65
, 012707
(2001
).6.
J. H.
Rose
, “Time reversal, focusing and exact inverse scattering
,” in Imaging of Complex Media with Acoustic and Seismic Waves
, edited by M.
Fink
, W. A.
Kuperman
, J. P.
Montagner
, and A.
Tourin
(Springer
, Berlin
, 2002
), pp. 97
–106
.7.
J. H.
Rose
, “ ‘Single-sided’ autofocusing of sound in layered materials
,” Inverse Probl.
18
, 1923
–1934
(2002
).8.
F.
Broggini
and R.
Snieder
, “Connection of scattering principles: A visual and mathematical tour
,” Eur. J. Phys.
33
, 593
–613
(2012
).9.
R. L.
Weaver
and O. I.
Lobkis
, “Ultrasonics without a source: Thermal fluctuation correlations at MHz frequencies
,” Phys. Rev. Lett.
87
, 134301
(2001
).10.
M.
Campillo
and A.
Paul
, “Long-range correlations in the diffuse seismic coda
,” Science
299
, 547
–549
(2003
).11.
K. G.
Sabra
, P.
Gerstoft
, P.
Roux
, W. A.
Kuperman
, and M. C.
Fehler
, “Surface wave tomography from microseisms in Southern California
,” Geophys. Res. Lett.
32
, L14311
, doi: (2005
).12.
D.
Draganov
, K.
Wapenaar
, W.
Mulder
, J.
Singer
, and A.
Verdel
, “Retrieval of reflections from seismic background-noise measurements
,” Geophys. Res. Lett.
34
, L04305
, doi: (2007
).13.
G. T.
Schuster
, “Theory of daylight/interferometric imaging: Tutorial
,” in EAGE, Extended Abstracts, A032
(2001
).14.
G. T.
Schuster
, Seismic Interferometry
(Cambridge University Press
, Cambridge
, 2009
), Chaps. 3–8.15.
A.
Bakulin
and R.
Calvert
, “The virtual source method: Theory and case study
,” Geophysics
71
, SI139
–SI150
(2006
).16.
A.
Curtis
, P.
Gerstoft
, H.
Sato
, R.
Snieder
, and K.
Wapenaar
, “Seismic interferometry–Turning noise into signal
,” The Leading Edge
25
, 1082
–1092
(2006
).17.
F.
Broggini
, R.
Snieder
, and K.
Wapenaar
, “Focusing the wavefield inside an unknown 1D medium: Beyond seismic interferometry
,” Geophysics
77
, A25
–A28
(2012
).18.
R. G.
Newton
, “Inverse scattering. IV. Three dimensions: Generalized Marchenko construction with bound states, and generalized Gel'fand-Levitan equations
,” J. Math. Phys.
23
, 594
–604
(1982
).19.
D. E.
Budreck
and J. H.
Rose
, “Three-dimensional inverse scattering in anisotropic elastic media
,” Inverse Probl.
6
, 331
–348
(1990
).20.
K.
Wapenaar
, J.
Thorbecke
, J.
van der Neut
, F.
Broggini
, E.
Slob
, and R.
Snieder
, “Marchenko imaging
,” Geophysics
79
(2014
) (to be published).21.
F.
Broggini
, R.
Snieder
, and K.
Wapenaar
, “Data-driven wave field focusing and imaging with multidimensional deconvolution: Numerical examples for reflection data with internal multiples
,” Geophysics
79
(2014
) (to be published).22.
J. R.
Berryhill
, “Wave-equation datuming before stack
,” Geophysics
49
, 2064
–2066
(1984
).23.
A. J.
Berkhout
and C. P. A.
Wapenaar
, “A unified approach to acoustical reflection imaging. Part II: The inverse problem
,” J. Acoust. Soc. Am.
93
, 2017
–2023
(1993
).24.
J. P.
Corones
, M. E.
Davison
, and R. J.
Krueger
, “Direct and inverse scattering in the time domain via invariant imbedding equations
,” J. Acoust. Soc. Am.
74
, 1535
–1541
(1983
).25.
L.
Fishman
, J. J.
McCoy
, and S. C.
Wales
, “Factorization and path integration of the Helmholtz equation: Numerical algorithms
,” J. Acoust. Soc. Am.
81
, 1355
–1376
(1987
).26.
C. P. A.
Wapenaar
and A. J.
Berkhout
, Elastic Wave Field Extrapolation
(Elsevier
, Amsterdam
, 1989
), Chaps. 3 and 5, App. B.27.
L.
Fishman
, “One-way propagation methods in direct and inverse scalar wave propagation modeling
,” Radio Sci.
28
, 865
–876
, doi: (1993
).28.
M. V.
de Hoop
, “Directional decomposition of transient acoustic wave fields
,” Ph.D. thesis, Delft University of Technology
(1992
), Chap. 5.29.
M. V.
de Hoop
, “Generalization of the Bremmer coupling series
,” J. Math. Phys.
37
, 3246
–3282
(1996
).30.
A. J.
Haines
and M. V.
de Hoop
, “An invariant imbedding analysis of general wave scattering problems
,” J. Math. Phys.
37
, 3854
–3881
(1996
).31.
C. P. A.
Wapenaar
, M. W. P.
Dillen
, and J. T.
Fokkema
, “Reciprocity theorems for electromagnetic or acoustic one-way wave fields in dissipative inhomogeneous media
,” Radio Sci.
36
, 851
–863
, doi: (2001
).32.
A. J.
Haines
, “Multi-source, multi-receiver synthetic seismograms for laterally heterogeneous media using F-K domain propagators
,” Geophys. J. Int.
95
, 237
–260
(1988
).33.
B. L. N.
Kennett
, K.
Koketsu
, and A. J.
Haines
, “Propagation invariants, reflection and transmission in anisotropic, laterally heterogeneous media
,” Geophys. J. Int.
103
, 95
–101
(1990
).34.
K.
Koketsu
, B. L. N.
Kennett
, and H.
Takenaka
, “2-D reflectivity method and synthetic seismograms for irregularly layered structures—II. Invariant embedding approach
,” Geophys. J. Int.
105
, 119
–130
(1991
).35.
H.
Takenaka
, B. L. N.
Kennett
, and K.
Koketsu
, “The integral operator representation of propagation invariants for elastic waves in irregularly layered media
,” Wave Motion
17
, 299
–317
(1993
).36.
C. P. A.
Wapenaar
, “One-way representations of seismic data
,” Geophys. J. Int.
127
, 178
–188
(1996
).37.
A. T.
de Hoop
, “Time-domain reciprocity theorems for acoustic wave fields in fluids with relaxation
,” J. Acoust. Soc. Am.
84
, 1877
–1882
(1988
).38.
J. T.
Fokkema
and P. M.
van den Berg
, Seismic Applications of Acoustic Reciprocity
(Elsevier
, Amsterdam
, 1993
), Chaps. 5 and 6.39.
K.
Wapenaar
, J.
Thorbecke
, and D.
Draganov
, “Relations between reflection and transmission responses of three-dimensional inhomogeneous media
,” Geophys. J. Int.
156
, 179
–194
(2004
).40.
K.
Wapenaar
, F.
Broggini
, E.
Slob
, and R.
Snieder
, “Three-dimensional single-sided Marchenko inverse scattering, data-driven focusing, Green's function retrieval, and their mutual relations
,” Phys. Rev. Lett.
110
, 084301
(2013
).41.
E.
Slob
, K.
Wapenaar
, F.
Broggini
, and R.
Snieder
, “Seismic reflector imaging using internal multiples with Marchenko-type equations
” Geophysics
79
(2
), S63
–S76
(2014
).42.
C. P. A.
Wapenaar
, G. L.
Peels
, V.
Budejicky
, and A. J.
Berkhout
, “Inverse extrapolation of primary seismic waves
,” Geophysics
54
, 853
–863
(1989
).43.
C.
Prada
, F.
Wu
, and M.
Fink
, “The iterative time reversal mirror: A solution to self-focusing in the pulse echo mode
,” J. Acoust. Soc. Am.
90
, 1119
–1129
(1991
).44.
G.
Montaldo
, J. F.
Aubry
, M.
Tanter
, and M.
Fink
, “Spatio-temporal coding in complex media for optimum beamforming: The iterative time-reversal approach
” IEEE Trans. Ultrason. Ferroelectr., Freq. Control
52
, 220
–230
(2005
).45.
M.
Fink
, “Time-reversal acoustics
,” J. Phys. Conf. Ser.
118
, 012001
(2008
).46.
D. J.
Verschuur
, A. J.
Berkhout
, and C. P. A.
Wapenaar
, “Adaptive surface-related multiple elimination
,” Geophysics
57
, 1166
–1177
(1992
).47.
G. J. A.
van Groenestijn
and D. J.
Verschuur
, “Estimation of primaries and near-offset reconstruction by sparse inversion: Marine data applications
,” Geophysics
74
, R119
–R128
(2009
).48.
J.
Virieux
and S.
Operto
, “An overview of full-waveform inversion in exploration geophysics
,” Geophysics
74
, WCC1
–WCC26
(2009
).49.
Q.
Liu
and Y. J.
Gu
, “Seismic imaging: From classical to adjoint tomography
,” Tectonophysics
566–567
, 31
–66
(2012
).50.
F.
Broggini
, R.
Snieder
, and K.
Wapenaar
, “Focusing inside an unknown medium using reflection data with internal multiples: numerical examples for a laterally-varying velocity model, spatially-extended virtual source, and inaccurate direct arrivals
,” in SEG, Expanded Abstracts, SPMUL 1.8
(2012
).51.
A. T.
de Hoop
, Handbook of Radiation and Scattering of Waves
(Academic Press
, London
, 1995
), Chap. 2.© 2014 Acoustical Society of America.
2014
Acoustical Society of America
You do not currently have access to this content.