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Interferometric Surface-Wave Isolation and Removal

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Interferometric Surface-Wave Isolation and Removal GEOPHYSICS, VOL.72, NO. 5 ͑SEPTEMBER-OCTOBER 2007͒; P.A69–A73, 4 FIGS. 10.1190/1.2761967 Interferometric surface-wave isolation and removal David F. Halliday1, Andrew Curtis2, Johan O. A. Robertsson3, and Dirk-Jan van Manen3 When using active sources, theory indicates that sources are re- ABSTRACT quired to form a surface that bounds the portion of the earth in which we are interested. In practice, this requirement can be relaxed. In the The removal of surface waves ͑ground roll͒ from land seis- virtual source method of Bakulin and Calvert ͑2004, 2006͒, for ex- mic data is critical in seismic processing because these waves ample, sources are located only at ͑or near͒ the earth’s surface and re- tend to mask informative body-wave arrivals. Removal be- ceivers are placed in a borehole. Recordings from these sources are comes difficult when surface waves are scattered, and data used to create virtual sources at the downhole receivers. quality is often impaired. We apply a method of seismic inter- In the related field of passive seismic interferometry, crosscorrela- ferometry, using both sources and receivers at the surface, to tions of ambient noise at periods of 5–20 s can produce estimates of estimate the surface-wave component of the Green’s function the ͑direct͒ Rayleigh wave between surface receivers ͑Shapiro and between any two points. These estimates are subtracted adap- Campillo, 2004; Shapiro et al., 2005͒. The authors argue that the tively from seismic survey data, providing a new method of Rayleigh-wave component is isolated in these crosscorrelations be- ground-roll removal that is not limited to nonscattering cause ͑a͒ Rayleigh waves dominate the Green’s function between regions. surface locations, and ͑b͒ near-surface noise sources in this relative- ly low-frequency band preferentially excite Rayleigh waves. We use a method similar to the virtual source approach but with a very different geometry, where both sources and receivers are locat- INTRODUCTION ed at the surface. Using the same reasoning as Shapiro et al. ͑2005͒, we expect ͑and observe͒ that our active source method produces in- In reflection seismology, surface waves constitute a strong source terreceiver signals that are dominated by surface waves. In a conven- of noise that is difficult to remove and often obscures the reflected tional land seismic survey, the response to surface sources can be re- waves in which we are interested. In heterogeneous media, removal corded explicitly at any planned source and receiver locations. This of these surface waves by conventional methods ͑such as f-k filter- allows the interferometric interreceiver surface waves to be re- ing͒ is often difficult because their energy is distributed over a wide moved from source-receiver survey data, providing a new method of portion of the f-k spectrum. We present a new method of surface- ground-roll removal ͑Curtis et al., 2006; Dong et al., 2006͒. Because wave removal that can be applied in either the time or frequency do- interferometry works in any degree of heterogeneity and is shown to main and has the potential to be effective in media with both homo- work better in more strongly scattering media ͑Larose et al., 2005͒, geneous and heterogeneous near-surface conditions. so should this new method. We can show that the Green’s function accounting for wave prop- agation between two points in lossless media can be synthesized by crosscorrelations of wavefield recordings made at each point. These GUIDED-WAVE CONSTRUCTION IN A wavefields are excited by distributed active or passive ͑noise͒ sourc- TWO-LAYER ACOUSTIC MODEL es. Details of the method differ, depending on the source type ͑Wap- enaar, 2004; van Manen et al., 2005, 2006; Wapenaar and Fokkema, Wapenaar and Fokkema ͑2006͒ use reciprocity theorems of 2006͒. Such methods have found several applications in seismology, the correlation type to show that the acoustic Green’s function ˆ where they are referred to collectively as seismic interferometry. G͑xB,xA,␻͒ between two points xB and xA can be determined by Manuscript received by the Editor January 31, 2007; revised manuscript received May 24, 2007; published onlineAugust 21, 2007. 1University of Edinburgh, Grant Institute of Earth Science, Edinburgh, Scotland, United Kingdom. E-mail: [email protected]. 2University of Edinburgh, Grant Institute of Earth Science, Edinburgh, Scotland, United Kingdom, and Edinburgh Collaborative of Subsurface Science and Engineering ͑ECOSSE͒. E-mail: [email protected]. 3WesternGeco London Technology Centre, Schlumberger House, Gatwick, United Kingdom. E-mail: [email protected]; dmanen @gatwick.westerngeco.slb.com. © 2007 Society of Exploration Geophysicists.All rights reserved. A69 A70 Halliday et al. waves ͑contributing predominantly upgoing waves͒ and can be ne- Gˆ *͑x ,x ,␻͒ + Gˆ ͑x ,x ,␻͒ B A B A glected. Furthermore, using simple geometric arguments and apply- −1 ing the stationary phase method of Snieder et al. ͑2006͒, we can ␻͒ ͑ ˆ ץͶ ͑ ˆ *͑ ␻͒ -D1 ͑a position providing a domiץ G xA,x, iG xB,x, show that for a stationary point on = j␻␳͑x͒ ץ D1 nant contribution to the interferometric integral͒, an equivalent sta- ͑ ͒ ␻͒͒ ˆ ͑ ␻͒͒ 2 ͑ ͒ tionary point exists at the surface Figure 1b and c . This allows us to ͑* ˆ ץ͑ − iG xA,x, G xB,x, nid x, 1 D1 in equation 2 by a summation over Nץ replace the integration over where j = ͑−1͒0.5, ␻ is angular frequency, ␳͑x͒ is density at x, surface source positions xn: ˆ G͑xA,x,␻͒ is pressure at xA arising from a monopole source at x, and ˆ Gˆ *͑x ,x ,␻͒ + Gˆ ͑x ,x ,␻͒ iG͑xA,x,␻͒ is pressure at xA from a dipole source at x. Here, the as- B A B Aץ terisk ͑ * ͒ denotes complex conjugation ͑corresponding to time-re- N versal in the time domain͒ and ni is the outward normal to integration ˆ *͑ ␻͒ ˆ ͑ ␻͒ ͑ ͒ = C ͚ G xA,xn, G xB,xn, , 3 D1, where this boundary encloses the locations xA and xB n =1ץ boundary D1 need not span the free surface because the integrand where C is a scaling factor. Note that despite source points being atץ ,in this case͓ is zero there͔. the free surface, this expression is nonzero because there are no de- By a simple process of crosscorrelation and integration, equation rivatives present ͑c.f., equation 1͒. 1 determines the Green’s function between two receivers. Similar Equation 3 is similar to equation 1 in Bakulin and Calvert ͑2004͒ expressions exist for differing source and receiver types and for elas- but with a very different geometry using both surface sources and tic media ͑van Manen et al., 2005, 2006; Wapenaar and Fokkema, surface receivers. The results of Bakulin and Calvert ͑2004, 2006͒ 2006͒. show that despite the assumptions involved, expressions such as Equation 1 requires both monopole and dipole sources; hence, it is equation 3 can be effective when using real data, where the boundary difficult to apply in practice. To simplify this expression, Wapenaar of sources is neither spherical nor located within a homogeneous re- .D1 is homoge- gionץ and Fokkema ͑2006͒ assume that the region outside neous ͑with propagation velocity c and mass density ␳͒ and that the Consider a group of multiples from a subsurface source that inter- boundary is a sphere with large radius. These assumptions lead to the fere to form a guided wave within the upper layer of Figure 1a. The expression acoustic guided wave is formed in a process similar to that of elastic surface waves ͑Shearer, 1999͒. From Figure 1b and c, we can see that Gˆ *͑x ,x ,␻͒ + Gˆ ͑x ,x ,␻͒ B A B A stationary points exist at the surface for any order of reflection, simi- 2 lar to the findings of Sabra et al. ͑2005͒, contributing multiples ͑and Ͷ ˆ *͑ ␻͒ ˆ ͑ ␻͒ 2 ͑ ͒ ͒ = G xA,x, G xB,x, d x, 2 hence guided waves to equation 3. Therefore, an array of surface ץ ␳c D1 sources located in the region where we expect these guided-wave where only monopole sources are required. True amplitudes may not stationary points to occur could be used to stimulate the guided- be estimated correctly where these assumptions are not met, but the wave component of the Green’s function. Mehta et al. ͑2007͒ show ͑ phase can still be recovered correctly Korneev and Bakulin, 2006; that upgoing reflections provide a weak contribution when com- ͒ Wapenaar and Fokkema, 2006 . This expression requires sources pared to the contribution of the direct wave ͑or guided wave, in this that surround the medium of interest, which is impractical, so we case͒. However, to ensure that any primary reflected-wave contribu- now further approximate equation 2. tion to equation 3 is further minimized, we avoid placing sources Consider the two-layer model of Figure 1a, of which the integra- near the locations where we expect primary reflection stationary ץ tion boundary D1 encloses a part. We assume that the deepest sourc- points. es provide a relatively small contribution to the interreceiver surface To demonstrate our method, we implement equation 3 in the sim- ple 2D model shown in Figure 1a. Data are computed using a vis- a) 600 m coelastic finite-difference code ͑Robertsson et al., 1994 ͒, with both A B attenuation and S-wave motion disabled. We first compute the 1 = 1000 m/s 20 m 3 Green’s function between points A and B ͑located just below the free 1 = 1300 kg/m surface͒, band-limiting the result with a 25-Hz Ricker wavelet.
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