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Journal ArticleDOI

Focusing waves in an unknown medium without wavefield decomposition

11 May 2021-Vol. 1, Iss: 5, pp 055602
TL;DR: In this article, the Marchenko equation for focusing without wavefield decomposition is derived, and by iteratively solving the MCE, the Green's function for an arbitrary location in the medium is retrieved from the scattered waves recorded on a closed receiver array and an estimate of the direct-wave without wave-field decompositions.
Abstract: The Gel'fand-Levitan equation, the Gopinath-Sondhi equation, and the Marchenko equation are developed for one-dimensional inverse scattering problems. Recently, a version of the Marchenko equation based on wavefield decomposition has been introduced for focusing waves in multi dimensions. However, wavefield decomposition is a limitation when waves propagate horizontally at the focusing level. Here, the Marchenko equation for focusing without wavefield decomposition is derived, and by iteratively solving the Marchenko equation, the Green's function for an arbitrary location in the medium is retrieved from the scattered waves recorded on a closed receiver array and an estimate of the direct-wave without wavefield decomposition.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors present a new derivation of Green's function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation, and these representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.
Abstract: Marchenko methods are based on integral representations which express Green’s functions for virtual sources and/or receivers in the subsurface in terms of the reflection response at the surface. An underlying assumption is that inside the medium the wave field can be decomposed into downgoing and upgoing waves and that evanescent waves can be neglected. We present a new derivation of Green’s function representations which circumvents these assumptions, both for the acoustic and the elastodynamic situation. These representations form the basis for research into new Marchenko methods which have the potential to handle refracted and evanescent waves and to more accurately image steeply dipping reflectors.

15 citations

Journal ArticleDOI
01 Jan 2022
TL;DR: In this paper , a unified matrix-vector wave equation for different wave phenomena is reformulated in terms of Green's matrices, source vectors, and wave field vectors, which makes these representations a suitable basis for developing advanced inverse scattering, imaging and monitoring methods for wave fields acquired on a single boundary.
Abstract: Classical acoustic wave-field representations consist of volume and boundary integrals, of which the integrands contain specific combinations of Green's functions, source distributions, and wave fields. Using a unified matrix-vector wave equation for different wave phenomena, these representations can be reformulated in terms of Green's matrices, source vectors, and wave-field vectors. The matrix-vector formalism also allows the formulation of representations in which propagator matrices replace the Green's matrices. These propagator matrices, in turn, can be expressed in terms of Marchenko-type focusing functions. An advantage of the representations with propagator matrices and focusing functions is that the boundary integrals in these representations are limited to a single open boundary. This makes these representations a suitable basis for developing advanced inverse scattering, imaging and monitoring methods for wave fields acquired on a single boundary.

3 citations

Journal ArticleDOI
TL;DR: In this article , the authors extend the concept of wavefield focusing by using a generalised homogeneous Green's function, which is based on partial differential equations and thus allows for additional insights and a new physical intuition for Marchenko equations.

3 citations

Proceedings ArticleDOI
15 Aug 2022
TL;DR: Wapenaar et al. as mentioned in this paper generalize the relation between the propagator matrix and the Marchenko focusing functions for a 3D inhomogeneous dissipative medium, and for the same type of medium, they discuss a relation between transfer matrix and Marchenko focus functions.
Abstract: PreviousNext No AccessSecond International Meeting for Applied Geoscience & EnergyThe propagator and transfer matrix for a 3D inhomogeneous dissipative acoustic medium, expressed in Marchenko focusing functionsAuthors: Kees WapenaarSjoerd de RidderMarcin DukalskiChristian ReinickeKees WapenaarDelft University of TechnologySearch for more papers by this author, Sjoerd de RidderUniversity of LeedsSearch for more papers by this author, Marcin DukalskiAramco Delft Global Research CenterSearch for more papers by this author, and Christian ReinickeAramco Delft Global Research CenterSearch for more papers by this authorhttps://doi.org/10.1190/image2022-3735927.1 SectionsAboutPDF/ePub ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InRedditEmail AbstractStandard Marchenko redatuming and imaging schemes neglect evanescent waves and are based on the assumption that decomposition into downgoing and upgoing waves is possible in the subsurface. Recently we have shown that propagator matrices, which circumvent these assumptions, can be expressed in terms of Marchenko focusing functions. In this paper we generalize the relation between the propagator matrix and the Marchenko focusing functions for a 3D inhomogeneous dissipative medium. Moreover, for the same type of medium we discuss a relation between the transfer matrix and the Marchenko focusing functions.Keywords: Marchenko, multiple, propagatorPermalink: https://doi.org/10.1190/image2022-3735927.1FiguresReferencesRelatedDetails Second International Meeting for Applied Geoscience & EnergyISSN (print):1052-3812 ISSN (online):1949-4645Copyright: 2022 Pages: 3694 publication data© 2022 Published in electronic format with permission by the Society of Exploration Geophysicists and the American Association of Petroleum GeologistsPublisher:Society of Exploration Geophysicists HistoryPublished Online: 15 Aug 2022 CITATION INFORMATION Kees Wapenaar, Sjoerd de Ridder, Marcin Dukalski, and Christian Reinicke, (2022), "The propagator and transfer matrix for a 3D inhomogeneous dissipative acoustic medium, expressed in Marchenko focusing functions," SEG Technical Program Expanded Abstracts : 3141-3145. https://doi.org/10.1190/image2022-3735927.1 Plain-Language Summary KeywordsMarchenkomultiplepropagatorPDF DownloadLoading ...

2 citations

Journal ArticleDOI
TL;DR: In this paper , an auxiliary equation for the forward-scattered components of the initial focusing function is derived based on these transmission data, and this equation can be solved in an acoustic medium with mass density contrast and constant propagation velocity.
Abstract: A Green's function in an acoustic medium can be retrieved from reflection data by solving a multidimensional Marchenko equation. This procedure requires a priori knowledge of the initial focusing function, which can be interpreted as the inverse of a transmitted wavefield as it would propagate through the medium, excluding (multiply) reflected waveforms. In practice, the initial focusing function is often replaced by a time-reversed direct wave, which is computed with help of a macro velocity model. Green's functions that are retrieved under this (direct-wave) approximation typically lack forward-scattered waveforms and their associated multiple reflections. We examine whether this problem can be mitigated by incorporating transmission data. Based on these transmission data, we derive an auxiliary equation for the forward-scattered components of the initial focusing function. We demonstrate that this equation can be solved in an acoustic medium with mass density contrast and constant propagation velocity. By solving the auxiliary and Marchenko equation successively, we can include forward-scattered waveforms in our Green's function estimates, as we demonstrate with a numerical example.

1 citations

References
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Journal ArticleDOI
24 Jan 2003-Science
TL;DR: This seismological example shows that diffuse waves produced by distant sources are sufficient to retrieve direct waves between two perfectly located points of observation and has potential applications in other fields.
Abstract: The late seismic coda may contain coherent information about the elastic response of Earth. We computed the correlations of the seismic codas of 101 distant earthquakes recorded at stations that were tens of kilometers apart. By stacking cross-correlation functions of codas, we found a low-frequency coherent part in the diffuse field. The extracted pulses have the polarization characteristics and group velocities expected for Rayleigh and Love waves. The set of cross-correlations has the symmetries of the surface-wave part of the Green tensor. This seismological example shows that diffuse waves produced by distant sources are sufficient to retrieve direct waves between two perfectly located points of observation. Because it relies on general properties of diffuse waves, this result has potential applications in other fields.

1,139 citations


"Focusing waves in an unknown medium..." refers methods in this paper

  • ...…retrieves the Green’s function for t> 0 for the virtual source location xs. Unlike other (interferometric) Green’s function retrieval methods (Campillo and Paul, 2003; Duroux et al., 2010; Roux et al., 2004; Sabra et al., 2005; Schuster, 2009; Snieder and Larose, 2013; Wapenaar et al.,…...

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Journal ArticleDOI
TL;DR: It is argued that acoustic thermal fluctuations, with displacement amplitudes of 3 fm, contain substantial ultrasonic information and it is shown that the noise autocorrelation function is the waveform that would be obtained in a direct pulse/echo measurement.
Abstract: Noise generated in an ultrasonic receiver circuit consisting of transducer and amplifier is usually ignored, or treated as a nuisance. Here it is argued that acoustic thermal fluctuations, with displacement amplitudes of 3 fm, contain substantial ultrasonic information. It is shown that the noise autocorrelation function is the waveform that would be obtained in a direct pulse/echo measurement. That thesis is demonstrated in experiments in which direct measurements are compared to correlation functions. The thermal nature of the elastodynamic noise that generates these correlations is confirmed by an absolute measurement of their strength, essentially a measurement of the sample temperature.

611 citations


"Focusing waves in an unknown medium..." refers methods in this paper

  • ...…(Campillo and Paul, 2003; Duroux et al., 2010; Roux et al., 2004; Sabra et al., 2005; Schuster, 2009; Snieder and Larose, 2013; Wapenaar et al., 2005; Weaver and Lobkis, 2001), no physical receiver is required at the position of the virtual source, and unlike other Marchenko methods (Wapenaar et…...

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Journal ArticleDOI
TL;DR: In this paper, an estimate of the Green's function between two seismic stations can be obtained from the time-derivative of the long-time average cross correlation of ambient noise between these two stations.
Abstract: [1] It has been demonstrated experimentally and theoretically that an estimate of the Green's function between two seismic stations can be obtained from the time-derivative of the long-time average cross correlation of ambient noise between these two stations. This TDGF estimate from just the noise field includes all tensor components of the Green's function and these Green's function estimates can be used to infer Earth structure. We have computed cross correlations using 1 to 30 continuous days of ambient noise recorded by over 150 broadband seismic stations located in Southern California. The data processing yielded thousands of cross-correlation pairs, for receiver separations from 4–500 km, which clearly exhibit coherent broadband propagating dispersive wavetrains across frequency band 0.1–2 Hz.

527 citations


"Focusing waves in an unknown medium..." refers methods in this paper

  • ...…location xs. Unlike other (interferometric) Green’s function retrieval methods (Campillo and Paul, 2003; Duroux et al., 2010; Roux et al., 2004; Sabra et al., 2005; Schuster, 2009; Snieder and Larose, 2013; Wapenaar et al., 2005; Weaver and Lobkis, 2001), no physical receiver is required at…...

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Journal ArticleDOI
TL;DR: In this paper, a method to obtain coherent acoustic wave fronts by measuring the space-time correlation function of ocean noise between two hydrophones is experimentally demonstrated, which exhibits deterministic waveguide arrival structure embedded in the time-domain Green's function.
Abstract: A method to obtain coherent acoustic wave fronts by measuring the space–time correlation function of ocean noise between two hydrophones is experimentally demonstrated. Though the sources of ocean noise are uncorrelated, the time-averaged noise correlation function exhibits deterministic waveguide arrival structure embedded in the time-domain Green’s function. A theoretical approach is derived for both volume and surface noise sources. Shipping noise is also investigated and simulated results are presented in deep or shallow water configurations. The data of opportunity used to demonstrate the extraction of wave fronts from ocean noise were taken from the synchronized vertical receive arrays used in the frame of the North Pacific Laboratory (NPAL) during time intervals when no source was transmitting.

283 citations


"Focusing waves in an unknown medium..." refers methods in this paper

  • ...…0 for the virtual source location xs. Unlike other (interferometric) Green’s function retrieval methods (Campillo and Paul, 2003; Duroux et al., 2010; Roux et al., 2004; Sabra et al., 2005; Schuster, 2009; Snieder and Larose, 2013; Wapenaar et al., 2005; Weaver and Lobkis, 2001), no physical…...

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Journal ArticleDOI
TL;DR: In this article, the Gelfand-Levitan integral equations are derived in the time domain and a new integral equation, similar to the Marchenko integral equation is also derived, which is used by Gopinath and Sondhi as a means of solving a time-dependent inverse problem arising in speech synthesis.

188 citations


"Focusing waves in an unknown medium..." refers methods in this paper

  • ...Burridge (1980) shows that the Gel’fand-Levitan equation and the Gopinath-Sondhi equation have the same structure as the Marchenko equation, and shows that the Marchenko equation can be used for medium reconstruction (Burridge, 1980; Newton, 1980a)....

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  • ...(3), we obtain Kðn̂; tÞ þ þ Aðn̂; n̂0; t þ tdðn̂0ÞÞdn0 þ þ ðt d t d Aðn̂; n̂0; t sÞKðn̂0; sÞ dsdn0 ¼ 0: (6) Burridge (1980) shows that the 1D Marchenko equation, Gel’fand-Levitan equation, and the Gopinath-Sondhi equations of inverse scattering can be written in symbolic notation as K þ Rþ Ð WRK ¼…...

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  • ...(6) has the same structure as the equations derived by Burridge (1980) and, therefore, gives a 2D Marchenko equation without using up/down decomposition....

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