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Tutorial on Seismic Interferometry: Part 2 — Underlying Theory and New Advances GEOPHYSICS, VOL.75, NO. 5 ͑SEPTEMBER-OCTOBER 2010͒; P.75A211–75A227, 10 FIGS. 10.1190/1.3463440 Tutorial on seismic interferometry: Part 2 — Underlying theory and new advances Kees Wapenaar1, Evert Slob1, Roel Snieder2, and Andrew Curtis3 ABSTRACT ent sources, gives the Green’s function emitted by a virtual source at the position of one of the receivers and observed by the In the 1990s, the method of time-reversed acoustics was devel- other receiver. This Green’s function representation for seismic oped. This method exploits the fact that the acoustic wave equa- interferometry is based on the assumption that the medium is tion for a lossless medium is invariant for time reversal. When ul- lossless and nonmoving. Recent developments, circumventing trasonic responses recorded by piezoelectric transducers are re- these assumptions, include interferometric representations for versed in time and fed simultaneously as source signals to the attenuating and/or moving media, as well as unified representa- transducers, they focus at the position of the original source, even tions for waves and diffusion phenomena, bending waves, quan- when the medium is very complex. In seismic interferometry the tum mechanical scattering, potential fields, elastodynamic, elec- time-reversed responses are not physically sent into the earth, but tromagnetic, poroelastic, and electroseismic waves. Significant they are convolved with other measured responses. The effect is improvements in the quality of the retrieved Green’s functions essentially the same: The time-reversed signals focus and create have been obtained with interferometry by deconvolution. A a virtual source which radiates waves into the medium that are trace-by-trace deconvolution process compensates for complex subsequently recorded by receivers. A mathematical derivation, source functions and the attenuation of the medium. Interferome- based on reciprocity theory, formalizes this principle: The cross- try by multidimensional deconvolution also compensates for the correlation of responses at two receivers, integrated over differ- effects of one-sided and/or irregular illumination. INTRODUCTION approaches that account for attenuating and/or nonreciprocal media, methods for obtaining virtual receivers or virtual reflectors, the rela- In Part 1, we discussed the basic principles of seismic interferom- tionship with imaging theory, and, last but not least, interferometry 4 etry ͑also known as Green’s function retrieval ͒ using mainly heuris- by deconvolution. The discussion of each of these advances is neces- tic arguments. In Part 2, we continue our discussion, starting with an sarily brief, but we include many references for further reading. analysis of the relation between seismic interferometry and the field of time-reversed acoustics, pioneered by Fink ͑1992, 1997͒. This analysis includes a heuristic discussion of the virtual-source method of Bakulin and Calvert ͑2004, 2006͒ and a review of an elegant phys- INTERFEROMETRY AND TIME-REVERSED ical derivation by Derode et al. ͑2003a, b͒ of Green’s function re- ACOUSTICS trieval by crosscorrelation.After that, we review exact Green’s func- Review of time-reversed acoustics tion representations for seismic interferometry in arbitrary inhomo- geneous, anisotropic lossless solids ͑Wapenaar, 2004͒ and discuss In the early 1990s, Mathias Fink and coworkers at the University the approximations that lead to the commonly used expressions. We of Paris VII initiated a new field of research called time-reversed conclude with an overview of recent and new advances, including acoustics ͑Fink, 1992, 1997; Derode et al., 1995; Draeger and Fink, Manuscript received by the Editor 30 November 2009; published online 14 September 2010. 1Delft University of Technology, Department of Geotechnology, Delft, The Netherlands. E-mail: [email protected]; [email protected]. 2Colorado School of Mines, Center for Wave Phenomena, Golden, Colorado, U.S.A. E-mail: [email protected]. 3University of Edinburgh, School of GeoSciences, Edinburgh, U.K. E-mail: [email protected]. © 2010 Society of Exploration Geophysicists.All rights reserved. 4By “Green’s function,” we mean the response of an impulsive point source in the actual medium rather than in a background medium. 75A211 Downloaded 15 Sep 2010 to 138.67.20.77. Redistribution subject to SEG license or copyright; see Terms of Use at http://segdl.org/ 75A212 Wapenaar et al. 1999; Fink and Prada, 2001͒. Here, we briefly review this research positions causes the particle to follow a completely different trajec- field; in the next sections, we discuss the links with seismic interfer- tory after only a few encounters with the scatterers. Waves, on the ometry. other hand, have a finite wavelength and travel along all possible tra- Time-reversed acoustics makes use of the fact that the acoustic jectories, visiting all of the scatterers in all possible combinations. wave equation for a lossless acoustic medium is invariant under time Hence, a small perturbation in initial conditions or scatterer posi- reversal ͑because it only contains even-order time derivatives, i.e., tions has a much less dramatic effect for wave scattering than for par- zeroth and second order͒. This means that when u͑x,t͒ is a solution, ticle scattering. Consequently, wave-propagation experiments -t͒ is a solution as well. Figure 1 illustrates the principle in through a strongly scattering medium have a high degree of repeatמ,then u͑x the context of an ultrasonic experiment ͑Derode et al., 1995; Fink, ability. Combined with the invariance of the wave equation for time 2006͒. A piezoelectric source at A in Figure 1a emits a short pulse reversal, this explains the excellent reproduction of the source wave- ͑duration of 1 ␮s͒ that propagates through a highly scattering medi- field after back propagation through the scattering medium. um ͑a set of 2000 randomly distributed steel rods with a diameter of As a historical side note, the idea of emitting time-reversed sig- 0.8 mm͒. The transmitted wavefield is received by an array of piezo- nals into a system was proposed and implemented in the 1960s ͑Par- electric transducers at B. The received traces ͑three are shown in Fig- vulescu, 1961, 1995͒. This was a single-channel method, aiming to ure 1a͒ exhibit a long coda ͑Ͼ200 ␮s͒ because of multiple scatter- compress a complicated response at a detector ͑for example, in an ing between the rods. Next, the traces are reversed in time and simul- ocean waveguide͒ into a single pulse. The method was proposed as a taneously fed as source signals to the transducers at B ͑Figure 1b͒. fast alternative to digital crosscorrelation, which, with the comput- This time-reversed wavefield propagates through the scattering me- ers at that time, cost on the order of 10 days’ computation time per dium and focuses at the position of the original source. Figure 1c correlation for signal lengths typically considered in underwater shows the received signal at the original source position; the dura- acoustics ͑Stewart et al., 1965͒. tion is of the same order as the original signal ͑ϳ1 ␮s͒. Figure 1d Snieder et al. ͑2002͒ and Grêt et al. ͑2006͒ exploit the repeatability shows beam profiles around the source position ͑amplitudes mea- of acoustic experiments in a method they call coda wave interferom- sured along the x-axis denoted in Figure 1b͒. The narrow beam is the etry ͑here, interferometry is used in the classical sense͒. Because the result of this experiment ͑back propagation via the scattering medi- scattering coda is repeatable when an experiment is carried out twice um͒, whereas the wide beam was obtained when the steel rods were under the same circumstances, any change in the coda between two removed. experiments can be attributed to changes in the medium. As a result The resolution is impressive; at the time, the stability of this ex- of the relatively long duration of the coda, minor time-lapse changes periment amazed many researchers. From a numerical experiment, in, for example, the background velocity can be monitored with high one might expect such good reconstruction; however, when waves accuracy by coda wave interferometry. have scattered by tens to hundreds of scatterers in a real experiment, Apart from repeatability, another important aspect of time-re- the fact that the wavefield refocuses at the original source point is versed acoustics is its potential to image beyond the diffraction limit. fascinating. Snieder and Scales ͑1998͒ have analyzed this phenome- Consider again the time-reversal experiment in Figure 1. An impor- non in detail. In their analysis, they compare wave scattering with tant effect of the scattering medium between the source at A and the particle scattering. They show for their model that, whereas particles transducer array at B is a widening of the effective aperture angle. In behave chaotically after having encountered typically eight scatter- other words, waves that arrive at each receiver include energy from a ers, waves remain stable after 30 or more scatterers. The instability much wider range of take-off angles from the source location than of particle scattering is explained by the fact that particles follow a would be the case without scatterers. A consequence is that time-re- single trajectory.Asmall disturbance in initial conditions or scatterer versal experiments in strongly scattering media have so-called superresolution properties ͑de a) c) Rosny and Fink, 2002; Lerosey et al., 2007͒. Ha- 1 nafy et al. ͑2009͒ and Cao et al. ͑2008͒ used this 0.5 property in a seismic time-reversal method to lo- A 0 cate trapped miners accurately after a mine col- –50 –25 25 50 –0.5 lapse. –1 µ B Time ( s) An essential condition for the stability and (First step) Normalized amplitude high-resolution aspects of time-reversed acous- b) d) 0 tics is that the time-reversed waves propagate through the same physical medium as in the for- –5 ward experiment. Here, we see a link between –10 A time-reversed acoustics and seismic interferome- –15 try.
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