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Ocean Acoustic Interferometry

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Ocean acoustic interferometryLaura A. Brooksa!and Peter GerstoftMarine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, California 92093!Received 12 January 2007; revised 13 March 2007; accepted 16 March 2007"Ocean acoustic interferometry refers to an approach whereby signals recorded from a line of sourcesare used to infer the Green’s function between two receivers. An approximation of the time domainGreen’s function is obtained by summing, over all source positions !stacking", the cross-correlationsbetween the receivers. Within this paper a stationary phase argument is used to describe therelationship between the stacked cross-correlations from a line of vertical sources, located in thesame vertical plane as two receivers, and the Green’s function between the receivers. Theory andsimulations demonstrate the approach and are in agreement with those of a modal based approachpresented by others. Results indicate that the stacked cross-correlations can be directly related to theshaded Green’s function, so long as the modal continuum of any sediment layers is negligible.© 2007 Acoustical Society of America. #DOI: 10.1121/1.2723650$PACS number!s": 43.30.Yj, 43.30.Gv, 43.60.Ac #AIT$ Pages: 3377–3385I. INTRODUCTIONApproximation of the Green’s function between twopoints in both open and closed environments has been a sub-ject of interest over the last few years. Lobkis and Weaver1showed, both theoretically and experimentally, that theGreen’s function between two points can be determined fromtheir temporal cross-correlation within a diffuse ultrasonicfield. This concept was extended by Derode et al.2whoshowed that the Green’s function can be conditionally recov-ered in an open scattering medium. They concluded that al-though the Green’s function can be determined from cross-correlations from a single source within a lossless closedcavity in which it is assumed that the eigenmodes do notdegenerate, the Green’s function will only emerge fromcross-correlations within an open scattering medium if thecross-correlations are summed over a perfect time-reversalmirror. Wapenaar3and Van Manen et al.4demonstrated thatretrieval of the Green’s function through summed cross-correlations can also be achieved in an inhomogeneous me-dium. Sánchez-Sesma et al. presented the canonical problemof elastodynamic Green’s function retrieval in a homoge-neous medium5and in an infinite space with a cylinderinclusion.6They showed that the Fourier transform of themean cross-correlation of motion between two points is pro-portional to the imaginary part of the corresponding exactGreen’s tensor, provided that there is equipartitioning of en-ergy.Extraction of the Green’s function by cross-correlationfrom ultrasonic noise,7–9ambient noise in a homogeneousmedium,10ambient ocean acoustic noise,11–14seismicnoise,15–20helioseismic noise,21and even moon-seismicnoise22has been undertaken, and the governing concepts arenow quite well understood.Three papers are of particular relevance to this work.Using a modal approach, Roux and Fink23showed theoreti-cally in the frequency domain and through simulation thatthe Green’s function between two points in a waveguide canbe determined by summing the cross-correlations from a ver-tical line of sources that are located in the same plane as thetwo receivers, external to the two receivers. Sabra et al.12used a stationary phase argument to formulate the time do-main Green’s function for time-averaged surface generatedambient noise cross-correlation, where the sources weremodeled as a horizontal plane of point sources at a constantdepth. Also using the method of stationary phase, Snieder etal.24presented a formulation that suggests that seismic inter-ferometry using sources equally spaced along the surface ofa homogeneous medium, with a horizontal reflector and nofree surface, in the same vertical plane as two receivers lo-cated within said medium, can be used to determine theGreen’s function between the two receivers. They showedthat when multiple bottom reflectors are included, spuriousmultiples may occur due to the region of interest not beingcompletely enclosed by sources, and concluded that, theo-retically, these spurious multiples could be removed by in-cluding an additional set of sources below the reflectors.Within this paper, the relationship between cross-correlations from a vertical line of active sources to two re-ceivers within a waveguide, and the shaded time domainGreen’s function between said receivers, is explored. We callthis approach ocean acoustic interferometry !OAI", as it isrelated to classical and seismic interferometry,25where inter-ferometry refers to the determination of information from theinterference phenomena between pairs of signals. Themethod of stationary phase is applied to simple reflectiveenvironments, providing an alternative theoretical means ofdescribing and understanding the physics governing thecross-correlation of such a source configuration, and howthis can be used to extract an amplitude shaded time domainGreen’s function. This work is distinct from work presentedby Roux and Fink,23Sabra et al.,12and Snieder et al.,24inthat a stationary phase derivation is applied to a column ofsources in a waveguide. A detailed physical and mathemati-cal discussion of spurious arrivals obtained in connectionwith OAI is presented. Numerical simulations of these envi-a"Also at the School of Mechanical Engineering, University of Adelaide,Australia; electronic mail: [email protected]. Acoust. Soc. Am. 121 "6!, June 2007 © 2007 Acoustical Society of America 33770001-4966/2007/121"6!/3377/9/$23.00ronments support the theory. A refractive environment withmore realistic water column, sediment, and bottom param-eters is also analyzed through numerical simulations. Thesegeometrically simple scenarios are chosen as they allow forunderstanding of the results, but the underlying concepts areapplicable to complex environments. Both the unstackedcross-correlations as a function of depth and the stackedcross-correlations are analyzed. The effect of limiting thesources to the water column is discussed and it is shown thatthe accuracy of OAI increases if the source column is ex-tended through the sediment. The spurious arrivals obtainedhere are compared with those obtained by Sabra et al.12andSnieder et al.24The manifestations of these aberrations aredistinct in each case and these differences


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