Extracting the Green s function from the correlation of coda waves: A derivation based on stationary phase

Transcription

1 PHYSICAL REVIEW E 69, Extracting the Green function from the correlation of coda wave: A derivation baed on tationary phae Roel Snieder Center for Wave Phenomena and Department of Geophyic, Colorado School of Mine, Golden, Colorado , USA Received 3 May 23; revied manucript received 2 December 23; publihed 29 April 24 he Green function of wave that propagate between two receiver can be found by cro-correlating multiply cattered wave recorded at thee receiver. hi technique obviate the need for a ource at one of thee location, and i therefore called paive imaging. hi principle ha been explained by auming that the normal mode of the ytem are uncorrelated and that all carry the ame amount of energy equipartitioning. Here I preent an alternative derivation of paive imaging of the ballitic wave that i not baed on normal mode. he derivation i valid for calar wave in three dimenion, and for elatic urface wave. Paive imaging of the ballitic wave i baed on the detructive interference of wave radiated from catterer away from the receiver line, and the contructive interference of wave radiated from econdary ource near the receiver line. he derivation preented here how that the global requirement of the equipartitioning of normal mode can be relaxed to the local requirement that the cattered wave propagate on average iotropically near the receiver. DOI: 1.113/PhyRevE PACS number: 43.2.g, 91.3.f, 42.3.d I. INRODUCION Paive imaging i a technique wherein wave recorded at two receiver location are correlated to give the Green function that decribe the direct wave propagation between thee receiver. he tail of multiply cattered wave i called the coda, after the Latin word for tail. Coda wave are effective for monitoring temporal change in media 1,2. Uing coda wave to determine the Green function i ueful becaue it provide information on wave propagation between two point in pace without the need for a ource at either of thee two point. he Green function thu obtained can be ued to form an image of the medium. Paive imaging ha been ued in eimic exploration 3,4, helioeimology 5, and ultraonic with either an active ource 6 8 or thermal noie that excite the coda 9,1. Numerical experiment have hown that paive imaging can be ued both in cloed and in open ytem 11,12. Campillo and Paul 13 recently ued paive imaging in crutal eimology by retrieving the urface wave Green function between eimological tation within Mexico uing coda wave generated by earthquake along the wet coat of Mexico. he theoretical explanation offered in their work i baed on the aumption of equipartitioning of the Earth mode 6. hee mode can either be the normal mode of the Earth, or the urface wave mode that decribe the guided wave that propagate along the Earth urface. Suppoe one invoke the Earth normal mode. In the tudy of Campillo and Paul 13, record of the ground motion with a duration of about 6 were ued. It take about 11 for a P wave to propagate to the other ide of the Earth 14; for an S wave it take even longer, o, in their tudy, the time i too hort for the Earth normal mode to equilibrate. Invoking the urface wave mode, however, alo poe a conceptual problem. hee mode are guided wave, and they are not dicrete becaue they exit for every frequency. For any given frequency there i a dicrete et of allowable wave number 15,16. Furthermore, becaue of the hallow depth of mot earthquake, the fundamental Love and Rayleigh wave mode are uually mot trongly excited and, in regional eimology, there i no equipartitioning of energy among urface wave mode becaue the fundamental Love and Rayleigh wave mode uually carry more energy than the um of all higher mode 17,18. hi mean that both the Earth normal mode a well a the urface wave mode cannot be ued to explain the experiment of Campillo and Paul 13. hi doe not mean that the derivation of paive imaging baed on normal mode 6 i incorrect, but it doe imply that it i not alway applicable. o goal of thi work i to preent an alternative way to undertand why the correlation hidden in the coda provide the ballitic wave Green function between the receiver. In Sec. II, I illutrate thi with the implet cae of calar wave in a homogeneou medium having embedded catterer. In Sec. III, the reult are interpreted and the role of the cattering medium i elucidated. In Sec. IV, I extend the derivation to elatic urface wave in three dimenion 3D. he derivation preented here i not baed on normal mode; therefore, it i valid both for cloed and open ytem. II. PASSIVE IMAGING FOR SCALAR WAVES IN A 3D MEDIUM Conider two receiver that are eparated by a ditance R, a hown in Fig. 1. I ue a coordinate ytem with the origin choen at receiver 1 and with the poitive x axi in the direction of receiver 2. he receiver are placed in a medium with catterer that radiate calar wave. Apart from the catterer, the propagation velocity i aumed to be contant. he catterer act a econdary ource of ingly and multiply cattered wave; catterer number emit a ignal S (t) that i due to all the wave that impinge upon that catterer. he wave field at the two receiver can be written a a uperpoition of the wave radiated by the catterer /24/694/46618/$ he American Phyical Society

2 ROEL SNIEDER FIG. 1. Definition of the geometric variable for the wave that travel from catterer number to two receiver. p 1,2 t S t r 2 1,2 r c 1,2, where c i the wave peed. Becaue of the directionality of the radiation pattern, the wave form recorded at the two receiver from a given catterer are not necearily equal. A hown later, however, the main contribution to thi um come from catterer near the receiver line. he wave traveling from thee catterer to the two receiver propagate in the ame direction. herefore, the directionality of the radiated energy i irrelevant. he contant 1/4 in the 3D Green function i included in the definition of S (t). If the repone of the receiver depend on frequency, then the impule repone of the receiver can be included in the wave form S (t). In paive imaging one correlate the wave recorded at two receiver 6 over a time window of length : C p2 tp 1 tdt. 2 Inerting Eq. 1 into thi expreion give a double um, over all catterer C S ts, t r 1 r 2 c dt/r 1 r 2. Let the autocorrelation of the ignal S (t) be denoted by 1 3 quairandom ignal 2, eimic imaging with drill bit noie 21, and time revered imaging 22. he double um, in expreion 3 can be plit into a um over diagonal term and a um over cro term. I how in the Appendix that for a random medium, the enemble average of the cro term vanihe provided the dc component of the S (t) i equal to zero. In a ingle realization of the medium, however, the cro term are nonzero. I alo how in the Appendix that for a ingle ource event e.g. an earthquake the ratio of the cro term to the diagonal term i maller than 2/. When an average over N rc ource event i carried out, thi ratio i bounded by 2/N rc. hi mean that by averaging over time, and poibly over different ource event, the um of the cro term can be made arbitrarily mall by increaing the time interval and the number of ource event N rc. In the following I refer to thi type of averaging a time/event averaging. Note that in everal tudie of paive imaging, time/ event averaging a decribed here i the only type of averaging that i applied 5 7,9,1,13. In the following I aume that ufficient time/event averaging i carried out o that the cro term in the um 3 can be ignored. With the definition 4 thi reduce expreion 3 to C r 1 r 2 r c 1 r 2. 5 C Since the Fourier tranform of the cro correlation i equal to the power pectrum 5 i given in the frequency domain by C S 2 expir 2 r 1 /c r 1 r 2. 6 he power pectrum S () 2 doe not depend on the phae fluctuation of the cattered wave, but i doe depend on fluctuation in the amplitude. When the variation in the () power pectrum are uncorrelated with the phae expi(r 2 r () 1 )/c, then expir CS 2 2 r 1 /c r, 7 1 r 2 with PHYSICAL REVIEW E 69, C S ts tdt. 4 S 2 1 N S 2, 8 In general, thi function i peaked around. he width of thi peak i denoted by, in the jargon of tochatic procee thi time i equal to the correlation time of the random proce S (t). hi width may vary among the different ignal; if that i the cae, indicate the generic width. When the S (t) are impulive function of time, then i of the ame order of magnitude a the width of the S (t). When thee ignal are of a long duration with a quairandom phae, can be much maller that the duration of the ignal. hi property ha been uccefully employed in radar imaging 19, exploration eimology uing vibrator that emit a where N i the number of catterer. When there are many catterer per wavelength, the ummation over catterer ( ) can be replaced by a volume integration ( )ndv that i weighted by the catterer denity n that i defined a the number of catterer per unit volume. In thi approximation Eq. 7 i given by CS expir 2r 2 1 /c ndxdydz, 9 r 1 r 2 with the ditance r 1 and r 2 defined in Fig

3 EXRACING HE GREEN FUNCION FROM HE... PHYSICAL REVIEW E 69, FIG. 2. Definition of the geometric variable for the wave that travel from a catterer at location r to two receiver. he region of contructive interference i indicated by the haded region. he integration over the tranvere coordinate x and y can be evaluated with the tationary phae approximation 23,24. hi technique leave only the contribution of the point near the receiver line yz, for which the integrand i not ocillatory. In thi approximation c C2S 2 i Rxx ndx. e ikrxx 1 For catterer to the left of the receiver (x) the integrand i given by exp(ikr)/r, for catterer to the right of the receiver (xr) the integrand i equal to exp(ikr)/r, and for catterer between the receiver (xr) the integrand i given by exp(ik(r2x))/r2x. Becaue the latter integrand i ocillatory, the region xr give a ubdominant contribution to the integral of Eq. 1. Ignoring thi contribution give 2 C8 2 S i c eikr e ikr ndx 4R 4R R ndx. 11 he term exp(ikr)/4r i the Green function that account for the wave that propagate between the receiver; thi term come from the integration over x. he econd term exp(ikr)/4r, which come from the integration over xr, i the advanced Green function. he retarded Green function come from the wave that propagate from receiver 1 to receiver 2 and correlate for a poitive lag time, a hown in the top panel of Fig. 3. he preence of the advanced Green function i due to the wave that propagate from receiver 2 to receiver 1; thee wave correlate for a negative lag time, a hown in the bottom panel of Fig. 3. he factor 1/i, which correpond to an integration in the time domain, come from the tationary phae evaluation of the x and y integral. In other tudie i wa alo noted that the correlation 2 give the integral of the um of the retarded and the advanced Green function 6,7,12. Malcolm et al. 8 ue thi property experimentally a a diagnotic of the equipartitioning of energy. Each catterer near the receiver line give, after time/ event averaging, the ame contribution to the Green function. hi lead to the integral ndx and R ndx that FIG. 3. he wave that propagate toward the right correlate for a poitive lag time top panel, while the wave that are leftmoving correlate for a negative lag time bottom panel. multiply the retarded and the advanced Green function, repectively. If the catterer denity n decreae ufficiently fat toward infinity, thee integral are finite, but in general the integral are infinite. Furthermore, the cattering loe incurred during the propagation from the catterer to the receiver have not yet been taken into account. In practice thi limit the volume integral to a region of a few mean free path of the receiver. hee unatifactory apect are addreed in the next ection. III. WHICH GREEN S FUNCION IS RERIEVED? he infinite integral in Eq. 11 can be removed by conidering the phyic of paive imaging in more detail. he concluion of the previou ection i that the correlation of the wave recorded at the two receiver yield the Green function by a proce of contructive interference of the cattered wave that propagate along the receiver line. In a cattering medium, the catterer affect the wave in three way: i the direction of wave propagation i changed by the catterer, ii the velocity of a tranmitted wave i affected by catterer near the path of propagation, and iii a tranmitted wave attenuate becaue of cattering loe. In an enemble average, the lat two effect are decribed by an effective medium 25,26. In a ingle realization of a cattering medium, the catterer alo leave an imprint on the phae velocity and attenuation of a propagating wave. hi i illutrated in Fig. 4 which how the wave that have propagated through a circular region with iotropic point catterer 27. he wave in the abence of catterer are hown with the dahed line, while the wave in the preence of catterer are hown by the thin olid line. All the receiver are at the ame ditance from the ource, yet there are appreciable variation in the amplitude and the phae of the ballitic wave due to the variation in the number of catterer within the firt Frenel zone for each ource-receiver pair. For a given realization and ource-receiver pair, a ballitic wave propagate with a phae velocity c, and attenuate over a ditance d with a factor exp(d/2l). he attenuation length L i not necearily equal to the mean free path l of the effective medium 25,26 becaue L i defined for a given path in a ingle realization. hi principle can be taken into account in Eq. 1 by interpreting c a the phae velocity of the ballitic wave, and

4 ROEL SNIEDER PHYSICAL REVIEW E 69, FIG. 5. Definition of the unit vector ˆ and ˆ that define the radial and tranvere polarization, repectively. e R/2L C8 2 S 4R eikr e R/2L 4R. 12 he x integral contribute a factor L to the correlation. he lat two term give the retarded and advanced Green function for the ballitic wave that propagate between the receiver. he iue of the medium of propagation i alo of relevance for the derivation of paive imaging baed on normal mode 6. hat derivation ha an open quetion: the normal mode of which ytem hould be ued? he normal mode of the true ytem, which include the catterer, are by definition uncoupled; equipartitioning among thee mode therefore will not occur. he normal mode of a homogeneou ytem are coupled by the catterer, which may reult in equipartitioning of energy among the mode of the homogeneou model. However, thi raie the quetion which homogeneou ytem to ue? It i not clear from the derivation of Lobki and Weaver 6 from which ytem one obtain the Green function. If thi would be the Green function of a medium that take the cattering loe of the ballitic wave into account, then that medium i attenuating. In uch a medium the normal mode are not orthogonal and the theory of Lobki and Weaver 6 mut be generalized by uing adjoint mode 28. IV. SURFACE WAVES IN AN ELASIC MEDIUM FIG. 4. Wave recorded at twelve location at the edge of a circular region that contain iotropic point catterer 27. he clock indicate the receiver poition. Shown i the wave field in the abence of catterer dahed line, the complete wave field in the preence of catterer thin olid line, and the wave field computed by averaging the catterer within the firt Frenel zone for each receiver thick olid line. by multiplying the integrand with a factor exp(rx x)/2l that account for the cattering loe of the wave that travel to both receiver. For a contant catterer denity n, the x integral that correpond to thoe in Eq. 1 can be carried out to give 2 ncl i eikr Campillo and Paul 6 obtained the full urface wave Green tenor by correlating the direct product of the three component of the two receiver. In thi ection, I how that the treatment of the previou ection can be generalized to urface wave propagating in a layered elatic 3D medium with embedded catterer. he urface-wave Green tenor of a layered medium whoe propertie depend on the depth z only can be written in the frequency domain a G ij r,r m G m ij r,r. 13 he total urface wave Green tenor i expreed a a um over urface-wave mode m that include both Rayleigh wave and Love wave. he urface wave Green tenor of mode m in the far field i given by 29,3 m r,r p m i z,p m j *z, eik m R/4, 14 2 k mr G ij where R(xx ) 2 (yy ) 2 i the ditance between the point meaured in the horizontal plane, and k m i the horizontal wave number of mode m. he polarization vector p m (z,) depend on the depth z and the azimuth of the path between point r and r. he orientation of the polarization vector can be expreed into the unit vector ˆ and ˆ that point in the radial and tranvere direction, repectively, a defined in Fig. 5. For Love wave the polarization vector i related to the Love wave eigenfunction l 1 m (z) 29,31 by while for Rayleigh wave p m z,l 1 m zˆ, p m z,r 1 m zˆ ir 2 m zẑ, with r 1 m (z) and r 2 m (z) the radial- and vertical-component modal function of the Rayleigh wave 29,31. Following Ref. 29, the urface-wave mode are aumed to be normalized according to the following convention:

5 EXRACING HE GREEN FUNCION FROM HE... PHYSICAL REVIEW E 69, c m U m l1 m 2 dz4c m U m r1 m 2 r 2 m 2 dz1, 17 with c m and U m are the phae velocity and group velocity of mode m, repectively, and (z) the ma denity. When the two receiver record the three component of the ground motion, one can form the correlation tenor of all combination of component C ij u 2i tu 1 j tdt, 18 where u 2i, for example, i the i component of the diplacement recorded at receiver 2. he recorded diplacement can be written a a um over the urface wave radiated by the different catterer. By analogy with Eq. 1 the diplacement of the two receiver in the frequency domain i given by a double um over catterer and urface wave mode m u 1,2 m p m z 1,2, 1,2 eik m X 1,2 /4 k m X 1,2 S,m. 19 () In thi expreion X 1,2 i the horizontal ditance between () catterer and receiver 1 and 2, repectively, 1,2 i the azimuth of the correponding cattering path, and S (,m) () i the frequency pectrum of the radiation of mode m from catterer. Inerting thi expreion in the correlation 18 give a double um, over catterer. he cro term interfere after ufficient time/event averaging detructively and can be ignored. he reulting um ( ) can be approximated with the urface integral ( )ndxdy, where n i the catterer denity per unit urface area. aking thee tep give C ij dxdynp m i z 2, 2 p m j *z 1, 1 m,m eik m X 2 k m X 1 k m k m X 2 X 1 S m S m *, 2 where it i undertood that all quantitie in Eq. 19 that depend on the catterer now depend on the location x,y of the integration point. he integral over the tranvere coordinate y can be evaluated in the tationary phae approximation, thi give a contribution from catterer near the receiver line that i given by C ij 2 dx n p m i z 2,p m j *z 1, m,m with eik m xrk m x e i/4 k m k m k m xk m xr Sm S m *, if if k m xr k m x, k m xr k m x. 22 he azimuth aociated with the polarization vector in Eq. 21 i given by for x and by for xr; thee cae are denoted by the plu and minu ign, repectively. he integrand of Eq. 21 i ocillatory, except when k m k m. hi mean that the dominant contribution come from the term mm, for thi reaon the mode coupling term mm can be ignored. Furthermore, the integrand i ocillatory over the range xr, and the dominant contribution come from the region x and xr. hee approximation give 1 C ij m ik m dx n m pi z 2,p m j *z 1, eik m R/4 1 ik m dx n m pi z 2, R 2 k mr p m j *z 1, eik m R/4 Sm 2, 23 2 k mr with S m () 2 the average power pectrum of the radiated mode m. he firt term i due to right-going wave that are generated in the region x, the polarization vector correpond to the azimuth, which i indicated by the plu ign. he econd term i due to wave cattered from the area xr that move toward the left, their polarization vector correpond to the azimuth, which i indicated by the minu ign. Note that the tationary phae integration over the tranvere coordinate lead to the correct geometrical preading 1/k m R. When the econdary ource of the wave are confined to the vertical plane through the ource and receiver, the integral over the tranvere coordinate i abent. hi i the reaon why the geometrical preading i not correctly retrieved in the derivation of Roux and Fink 11. A comparion with Eq. 14 how that the firt term i equal to (c m /i)g m ij (r 2,r 1 ), while the econd one equal (c m /i)g m ji (r 1,r 2 )*. he correlation tenor i therefore given by C ij m c m G m ijr 2,r 1 dx n i G m ijr 1,r 2 i R dx n S m 2,

6 ROEL SNIEDER PHYSICAL REVIEW E 69, where the dagger denote the Hermitian conjugate. hi expreion contain infinite integral. Incorporating the attenuative propertie of the ballitic urface wave, a hown in Sec. III, give C ij m S m 2 nc m L m G m ijr 2,r 1 i G m ijr 1,r 2 i, 25 where L m i attenuation length of urface wave mode m, and where the Green function of each mode i undertood to contain an attenuation term exp(r/2l m ). hi expreion i imilar to the correponding reult 12 for calar wave in three dimenion. he correlation give the uperpoition of the Green function of the ballitic wave that propagate from receiver 1 to receiver 2 the firt term, and the ballitic wave Green function that propagate in the oppoite direction the lat term. Paive imaging with urface wave thu provide the uperpoition of the retarded and advanced urface wave Green function of the ballitic wave. V. CONCLUSION A hown in Eq. 11 and 25, the ballitic wave Green function can be obtained by a cro correlation of the wave form at two receiver. wo tep mut be taken to extract thi Green function from the correlation. Firt, the correlation in the frequency domain mut be multiplied by i/s() 2. he multiplication with i correpond in the time domain to a differentiation that undoe the integration ued in the cro correlation. he diviion by the power pectrum S() 2 correct for frequency-dependent factor in the cattering coefficient, the ource pectrum, and the receiver repone. For the cae of Eq. 11 for calar wave in 3D, the power pectrum can be obtained from the wave recorded at the receiver. For the correponding expreion 25 for urface wave in an elatic medium, each mode mut be corrected for the power pectrum of that mode. he cattering coefficient for urface wave mode trongly depend on the depth of the catterer 29, and on topography 32. For thi reaon the average power pectrum S m () 2 of the cattered urface wave mode may depend trongly on the mode number m. It i not clear how S m () 2 can be extracted from the recorded wave. In application in crutal eimology, the fundamental mode Love and Rayleigh wave uually dominate. he average power in the fundamental Rayleigh wave can be etimated from the vertical component. he power of the horizontal component can then be ued to infer the power in the fundamental Love wave. Without correcting for the power pectrum, the cro correlation may not give the correct frequency dependence of the Green function. he econd tep that mut be taken i due to the fact that the cro correlation of the wave recorded at two receiver give the uperpoition of the retarded and the advanced ballitic wave Green function. hee two contribution can be unraveled in the time domain by retricting the ignal to FIG. 6. he wave path of a reflected wave. he receiver are hown by olid circle. he dark gray area indicate the location of catterer that give a tationary contribution to the integration over catterer for the reflected wave. he open circle denote the image of receiver 1 upon reflection in the free urface, and the light gray area i the image of the catterer that contribute to the tationary phae olution for the reflected wave. poitive and negative time window, repectively 6. Phyically, the derivation hown here implie that in general the cattered wave recorded at the two receiver are uncorrelated, except for the wave radiated from catterer that are located near the receiver line. Paive imaging of the ballitic wave thu i baed on contructive interference olely of thoe cattered wave that propagate along the receiver line. Ultraound experiment with a finite aluminum ample how that the ballitic wave a well a wave that are reflected from boundarie are recontructed from paive imaging 6,9,1. he theory preented here doe not account for thee reflected wave. When a wave reflect off a plane boundary, a hown in Fig. 6, the cattering path from catterer located in the dark gray area interfere contructively. he theory preented here can be applied to thi problem by invoking an image receiver and image catterer a indicated by the open circle and light-gray area in Fig. 6. For a nonplanar boundary or an inhomogeneou reference medium one need to determine other tationary phae contribution to the integral over the catterer. hee contribution depend on the geometry of cattering path, and are not accounted for by the theory preented here. he equilibration of normal mode 6 provide a ufficient condition for contructing the Green function from the cro correlation of the wave recorded at two receiver. he derivation preented here how, however, that the equilibration of normal mode i not a neceary condition. In fact, the derivation preented here i equally valid for open ytem that do not poe normal mode. he derivation alo hold for cloed ytem that do poe normal mode at early time when the mode have not yet equilibrated. he derivation preented here i baed on the aumption that the cattered wave propagate iotropically in all direction. hi doe not imply that the cattering coefficient are iotropic; it mean that the net energy flux of the cattered wave i mall. Mathematically thi i expreed by the condition that the catterer denity n i contant in pace. hi implie a local condition on the iotropic propagation of cat

7 EXRACING HE GREEN FUNCION FROM HE... tered wave near the receiver, rather than the global requirement of the equilibration of normal mode. By uing the correlation that are hidden in the coda wave, the detructive interference of wave radiated from catterer away from the receiver line, and the contructive interference of cattered wave that propagate along the receiver line make paive imaging an effective technique for extracting the ballitic wave Green function between two point without uing a ource at either of thee point. ACKNOWLEDGMENS I am much indebted to Arnaud Derode who reviewed an earlier verion of thi manucript hi critical and contructive input ha been crucial. I alo appreciate the comment from and dicuion with Ken Larner, Matt Haney, John Scale, and Alion Malcolm. hi work wa upported by the NSF Grant No. EAR and by the ponor of the Conortium Project on Seimic Invere Method for Complex Structure at the Center for Wave Phenomena. APPENDIX: ESIMAION OF HE CROSS ERMS IN A SINGLE REALIZAION Since the cattered wave in a complex medium have a random character, I etimate the cro term in the um 3 for a random medium. In thi model, the catter are randomly located in the medium and the location of different catterer i uncorrelated. In the following, I aume that the dc component of the ignal vanihe. If thi i the cae, then S t. A1 In thi appendix the angled bracket indicate an enemble average. It i eential that the dc component of the cattered wave vanih; when the dc component i nonzero there i no detructive interference, and averaging over the catterer poition doe not give a vanihing mean ignal. he wave emitted by catter and are in the enemble average uncorrelated becaue the poition of thee catterer are uncorrelated. hi mean that S ts ts ts t for, A2 where expreion A1 i ued in the lat identity. Following the notation of expreion 4 we have for the diagonal term S ts tc tt. A3 Strictly peaking thi covariance may depend on the time t a well, becaue the time erie S (t) in not necearily tationary. hi can be incorporated by replacing expreion A3 by S (t)s (t)w(t)c (tt), where W(t) varie lowly with time compared to C (tt) and compared to the width of the employed time window. hi complication can be incorporated by replacing C (tt) byw(t)c (tt). Since thi doe not change the eence of the argument, thi i not included in the following. Let u conider the um 3 and ignore for the moment the geometrical preading term r 1 () r 2 (). hee term can be inerted at the end, but they do not change the eence of the argument. Aborbing the term (r 1 () r 2 () )/c into, expreion 3 become A4 where the firt term denote the diagonal term C D () and the econd term give the cro term C C (). Becaue of Eq. A2 the expectation value of the cro term vanihe: C C S ts tdt. A5 In a ingle realization, however, the cro term C C () iin general nonzero. I etimate it value by analyzing the variance C 2 C (). Uing expreion A4 thi variance i given by C 2 C S ts u ts t,uu S u tdtdt. A6 Since the different S (t) are uncorrelated expreion A2, the only term that give a nonzero contribution to Eq. A6 are the term u and u or the term u and u. hi give C 2 C S ts ts ts t S ts ts ts tdtdt. A7 With the definition A3 thi can be written a C ttc tt C C 2 PHYSICAL REVIEW E 69, C ttc ttdtdt. A8 Now let u etimate the order of magnitude of thi um. When there are N catterer contributing to thi um, then there are N(N1)N 2 term in the double um. Let the maximum of C (t) be given by C max. hi maximum may be different for the different C (t), if that i the cae then C max i the larget of all thee maxima. he width of the autocorrelation C (t) i indicated by. Each of the t integral in Eq. A8 then give a contribution that i maller 2 than C max. he remaining t integral give a contribution. hi implie that C 2 C 2N 2 2 C max. A

8 ROEL SNIEDER In order to ae the importance of the cro term, I compare thi with the mean of the diagonal term. hi mean i given by C D S 2 tdt C dt. Uing the ame etimate that led to Eq. A9 then give A1 C D C NC max, A11 becaue the autocorrelation attain it maximum for a zero time lag. With the etimate A9 thi give the following ratio of the tandard deviation of the cro term to the diagonal term: PHYSICAL REVIEW E 69, C C 2 1/2 C D 2. A12 Note that thi ratio doe not depend on the number of catterer. When in addition to an averaging over time, N rc ource event are ued, and when the ignal emitted by the catterer for different ource event are uncorrelated, the tandard deviation of the cro term increae with a factor N rc while the diagonal term are proportional to N rc,o that C C 2 1/2 C D 2 N rc. A13 1 R. Snieder, A. Grêt, H. Douma, and J. Scale, Science 295, M. L. Cowan, I. P. Jone, J. H. Page, and D. A. Weitz, Phy. Rev. E 65, J. N. Louie, Bull. Seimol. Soc. Am. 91, K. Wapenaar, D. Draganov, J. horbecke, and J. Fokkema, Geophy. J. Int. 156, J. E. Rickett and J. F. Claerbout, Sol. Phy. 192, O. I. Lobki and R. L. Weaver, J. Acout. Soc. Am. 11, A. Derode, E. Laroe, M. Campillo, and M. Fink, Appl. Phy. Lett. 83, A. Malcolm, J. A. Scale, and B. van iggelen unpublihed. 9 R. L. Weaver and O. I. Lobki, Phy. Rev. Lett. 87, R. Weaver and O. Lobki, Ultraonic 4, P. Roux and M. Fink, J. Acout. Soc. Am. 113, A. Derode, E. Laroe, M. anter, J. de Rony, A. ourin, M. Campillo, and M. Fink, J. Acout. Soc. Am. 113, M. Campillo and A. Paul, Science 299, S. Stein and M. Wyeion, An Introduction to Seimology, Earthquake, and Earth Structure Blackwell, Malden, MA, H. akeuchi and M. Saito, in Seimology: Surface Wave and Earth Ocillation, Vol. 11 of Method in Computational Phyic, edited by B. A. Bolt Academic Pre, New York, F. A. Dahlen and J. romp, heoretical Global Seimology Princeton Univerity Pre, Princeton, animoto, Geophy. J. R. Atron. Soc. 89, B. Romanowicz, in International Handbook of Earthquake and Engineering Seimology, edited by W. H. K. Lee, H. Kanamori, P. C. Jenning, and C. Kilinger Academic Pre, New York, 23, pp , Pt. B. 19 J. P. Nathanon, F. E. znd Reilly, and M. N. Cohen, Radar Deign Principle, 2nd ed. Scitech Publihing, Mendham, P. L. Goupillaud, Geophyic 41, J. W. Rector and B. P. Marion, Geophyic 56, A. Derode, A. ourin, and M. Fink, J. Appl. Phy. 85, C. M. Bender and S. A. Orzag, Advanced Mathematical Method for Scientit and Engineer McGraw-Hill, New York, N. Bleitein, Mathematical Method for Wave Phenomena Academic Pre, Orlando, U. Frich, in Probabilitic Method in Applied Mathematic, edited by A.. Bharucha-Reid Academic Pre, New York, 1968, pp P. Sheng, Introduction to Wave Scattering, Localization, and Meocopic Phenomena Academic Pre, San Diego, J. Groenenboom and R. Snieder, J. Acout. Soc. Am. 98, J. Park and F. Gilbert, J. Geophy. Re. 91, R. Snieder, Geophy. J. R. Atron. Soc. 84, R. Snieder, in Scattering and Invere Scattering in Pure and Applied Science, edited by R. Pike and P. C. Sabatier Academic Pre, San Diego, 22, Chap , pp K. Aki and P. G. Richard, Quantitative Seimology, 2nd ed. Univerity Science Book, Saualito, R. Snieder, Phy. Earth Planet. Inter. 44,