Visiting from institute of Wave and Information, Xi an Jiaotong University, Xi an , China. B x and gmnpq( x ) are 2D functions.

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1 Efficiet 3D illuiatio aalyi uig local expoetial fae Jia Mao* Ru-Sha Wu Modelig ad Iagig Laboatoy, IGPP, Uiveity of Califoia, Sata Cuz Viitig fo ititute of Wave ad Ifoatio, Xi a Jiaotog Uiveity, Xi a 749, Chia Suay Local lat tack i a ueful ethod i extactig agle doai ifoatio. Howeve, it athe expeive to ue i 3D illuiatio aalyi. Hee, we peet a efficiet ethod of diectioal illuiatio aalyi fo 3D cae uig local expoetial fae. To get the local agle ifoatio, the wave field i decopoed by D local expoetial fae, which fo a tight fae of edudacy 4 i thi 3D geoety, ipleeted by the cobiatio of local coie ad ie tafo. A the local coie/ie tafo have fat algoith, thi ethod povide a efficiet tool of illuiatio aalyi, which i uch fate tha foally ued local lat tack ethod. We calculated the diectioal illuiatio (DI ap ad the acquiitio dip epoe (ADR fo the 3D SEG/EAGE alt odel to deotate the validity of ou ethod. Itoductio Seiic illuiatio aalyi i a ueful tool that etiate the potetial detectig powe of a pecific acquiitio yte fo a give velocity tuctue of the ediu. Taditioally, illuiatio aalyi i baed o the ay tacig techique (Scheide, 999; Bea et al., ; Muedte et al., abc. The ay-baed ethod ca povide both the iteity ad diectio ifoatio caied i the wavefield. Howeve, the high fequecy ayptotic appoxiatio ad the igulaity poble of the ay theoy ay eveely liit it accuacy i coplex egio (Hoffa,. Full-wave fiite-diffeece ethod i widely ued fo wave popagatio iulatio. But it uually povide oly the total illuiatio, ad it too expeive fo pactical applicatio fo illuiatio aalyi. Recetly, techique to obtai the localized agle doai ifoatio fo the fequecy doai wave field baed o bealet decopoitio o local lat tack have bee developed ad applied to diectioal illuiatio aalyi (Wu ad Che,, 3, 6; Xie ad Wu, ; Xie et al., 6. The Gabo-Daubechie fae decopoitio i coplete but ot othogoal, ad theefoe ha edudacy i the epeetatio. Local lat tack fo wavefield decopoitio ae coputatio deadig. The local coie bai (LCB decopoitio, o the othe had, i othogoal. Fat algoith of local coie/ie tafo exit. Howeve, local coie bealet have alway two lobe i yety with epect to the vetical axi. Thi lack of uiquely defied diectio-localizatio pevet it ue fo diectioal illuiatio aalyi. To ovecoe thi hotcoig, a local expoetial fae (LEF ethod (Mao ad Wu, 7 have bee uccefully developed fo D diectioal illuiatio aalyi with high efficiecy. I thi pape, we exted thi ethod to the 3D cae. Fit we itoduce the D local expoetial fae (LEF ad it applicatio to wavefield decopoitio i 3D cae, ad the give a bief deciptio o how to calculate the diectioal eegy fluxe fo the wavefield ad apply the to the illuiatio aalyi. To deotate the potetial applicatio of thi ethod, illuiatio exaple ae calculated fo the 3D SEG/EAGE Salt odel. D local expoetial fae The local coie bai cotucted by Coiffa ad Meye (99 (ee alo Mallat, 999 ca be chaacteized by poitio x, the oial legth of widow L = x+ x, ad waveube idex ( c x x ψ ( x = B ( x co π +, ( L L whee B ( x i a bell fuctio which i ooth ad uppoted i the copact iteval [ x ε, x + ε ] fo x + ε x+ ε, with ε, ε a the left ad ight ovelappig adiu, epectively. I the ae ae, the local ie tafo ca be defied. I ode to have uiquely pecified diectio i wave popagatio, we exted the local expoetial fae (Mao ad Wu, 7 to the D cae a follow g( x = Bq ( x exp( iξpi ( x xq, ( LL whee = ( x, y q ( q + x, x q = ( x, yq ad p = ( ξ, η p π ( + ξ. B x ad g( x ae D fuctio. ξ = L π ( p + ad η p = ae the local waveube. The D L q LEF ca alo be witte a ( ++ g ( x, y, ( + g ( x, y ad ( + g ( x, y, ( g ( x, y, whee the upecipt ( ++, ( +, ( + ad ( deote the diffeet aziuthal diectio SEG Houto 9 Iteatioal Expoitio ad Aual Meetig 35

2 3D illuiatio aalyi uig local expoetial fae (fo the fou quadat i the D decopoitio pla. Take ( + g ( x, y a exaple g, ( ( ( ξ ( η ( ( ( + 4 ( x, y = B x Bq y exp i x x exp i p y yq LL q (3 The D LEF i a tight fae with edudacy 4, which ca be ipleeted by liea cobiatio of D local tigooetic bae. Wave field decopoitio ito local waveube doai The fequecy doai wave field uxyzω (,,, at depth z ca be decopoed ito local expoetial bealet with widow alog the x-y plae uxyz (,,, = u ˆz ( x, yq, ξ, ξp, g ( xy, q p (4 whee uˆ z( x, yq, ξ, ξp, i the coefficiet of the local expoetial fae decopoitio, located at pace widow ( x, y q ad waveube widow ( ξ, ξ p. The coefficiet coepodig to local expoetial bealet ca be calculated by fat local coie/ie tafo. Figue. Defiitio of of icidet agle = (, ϕ cae fo 3D Fo the local plae wave of local hoizotal waveube vecto ξ = ( ξ, ξ p, the coepodig popagatig agle i ξ + ξ p ξ = (, aci, acta ϕp = k ξ p (5 Whee ad ϕ p ae the local icidet agle with epect to the z-axi ad x-axi epectively (how i Figue. i fo to 9, ad ϕ p i fo k = ω v the velocity at (x, z, whee x=(x, y. 8 to 8. i the local waveube ad v ( x, z (, z Illuiatio aalyi i the local agle doai x i Fo a give acquiitio geoety, the fequecy-pace Gee fuctio fo ouce to ubuface poit (x, z ca be decopoed i the iage egio to a uatio of local agle copoet G( x, z,, = G( x, z,,,, (6 whee G(, z,,,. x i it local-agle-copoet of Siilaly, the fequecy-pace Gee fuctio fo ubuface poit (x, z to eceive ca be decopoed a G( x, z,, = G( x, z,,,, (7 whee G(, z,,, x i it local-agle-copoet of. The diectioal illuiatio (DI eegy ditibutio i local agle doai i a uatio of the cotibutio fo all the ouce at the iage poit fo each local agle D (,,, (,,,, a x z ω = G x z ω. (8 I ode to evaluate the apetue ad popagatio effect of the give acquiitio geoety o eegy ditibutio fo a pecific pai of icidet/eceivig agle, we ue uit ipule a ouce at both ouce ad eceive poit fo the etie acquiitio cofiguatio ad aue a uit catteig coefficiet at each pace poit. Siila to the pocedue of DI appig, we u up cotibutio of the Gee fuctio fo each icidet/eceivig pai to get the acquiitio apetue efficacy (AAE atix at each iage poit, which eglect the detailed wave iteface patte ad coide oly the eegy ditibutio i pace ad agle of the acquiitio cofiguatio. The the AAE atix at poit (x, z i defied a (,,,, (,,,, (,,,, g ω = ω g ω E x z G x z G x z. (9 We ca futhe educe the acquiitio efficacy atix at each poit to a acquiitio dip epoe (ADR vecto which i a fuctio of eflecto dip oly. To get ADR, we fit tafe the ouce-eceive agle pai to the dipeflectio agle pai by a coodiate otatio. The we u up all the eflected eegy fo each eflecto-dip. Ad( x, z,, = E( x, z,,,, ( whee A (, z,, ad d x i the ADR vecto fo poit (x, z, i the aued eflecto oal agle (ae a SEG Houto 9 Iteatioal Expoitio ad Aual Meetig 36

3D illuiatio aalyi uig local expoetial fae igatio-dip agle (with epect to z-axi ad i the

The value of the ADR ap eaue the dip-agle epoe of the acquiitio yte, icludig the ouce ad

I ou exaple, we calculate oly the ADR fo the cetal fequecy ad ake the algoith vey

Vetical ad hoizotal lice of velocity odel ad the coepodig petack depth igatio iage Figue

which 3, ϕ 9 =, ϕ ad ae ( = =, ( ( 3, ϕ 9 = = epectively.

illuiatio aalyi, we calculated a ube of ueical exaple uig the 3D SEG/EAGE alt velocity

Thi epeet a lad type acquiitio geoety. The locatio of thoe 45 hot ae how i the Figue.

iage. The locatio of the lice ae aked i Figue. (e (f Figue 4.

3 3D illuiatio aalyi uig local expoetial fae igatio-dip agle (with epect to z-axi ad i the eflectio agle with epect to the oal. The value of the ADR ap eaue the dip-agle epoe of the acquiitio yte, icludig the ouce ad eceive apetue, ad popagatio effect. I ou exaple, we calculate oly the ADR fo the cetal fequecy ad ake the algoith vey efficiet. Nueical exaple Figue 3. Vetical ad hoizotal lice of velocity odel ad the coepodig petack depth igatio iage Figue 4 i the eult of DI ap with diffeet icidet agle fo the give vetical ad hoizotal lice, which 3, ϕ 9 =, ϕ ad ae ( = =, ( ( 3, ϕ 9 = = epectively. Fo the DI ap we ca ee the eegy flux of diffeet local agle clealy, epecially fo the ubalt aea. Figue. locatio of the 45 hot locatio of the lice fo copaio To deotate the applicatio of the illuiatio aalyi, we calculated a ube of ueical exaple uig the 3D SEG/EAGE alt velocity odel. Thi exaple iulate the illuiatio coditio of the 45 hot data et. Thi epeet a lad type acquiitio geoety. The locatio of thoe 45 hot ae how i the Figue. Figue 3 how the vetical ad hoizotal lice of the velocity odel ad petack depth igatio iage. The locatio of the lice ae aked i Figue. (e (f Figue 4. Diectioal illuiatio (DI ap uig LEF: = 3, ϕ = 9 ; DI fo agle DI fo agle ( ( =, ϕ ; (e(f DI fo agle ( = 3, ϕ = 9 Fo ADR calculatio, to get each eceive Gee fuctio eed a lage aout of coputatio ad toage, ad the we calculate oly oe eult of patially itegated ove cetai agle age. Figue 5 how the ADR ap calculated by the LEF ethod, the dip agle ae 9, ϕ 9 =, ϕ ad ( < < = ( SEG Houto 9 Iteatioal Expoitio ad Aual Meetig 37

3D illuiatio aalyi uig local expoetial fae ( 9, ϕ 9 < < =, that epeet egative, zeo ad poitive dip epoe epectively.

Fo figue 4, we ee that the illuiatio with egative aziuthal agle i uch pooe tha that of poitive aziuthal agle.

Fo thee illuiatio eult, we ca alo explai the poo iage quality of the ubalt aea fo eflecto with cetai dip agle. Lwi coputatio with O Lwi log 4 (due to the ue of foldig techique i fat LCT ad LST.

4 3D illuiatio aalyi uig local expoetial fae ( 9, ϕ 9 < < =, that epeet egative, zeo ad poitive dip epoe epectively. Thee figue how the diffeet epoe fo diffeet dip agle, which i due to the alt tuctue ad the acquiitio yte. Fo figue 4, we ee that the illuiatio with egative aziuthal agle i uch pooe tha that of poitive aziuthal agle. Coepodigly, i figue 5, the dip epoe fo eflecto with egative dip ae weake tha that with poitive dip. Fo thee illuiatio eult, we ca alo explai the poo iage quality of the ubalt aea fo eflecto with cetai dip agle. Lwi coputatio with O Lwi log 4 (due to the ue of foldig techique i fat LCT ad LST. Theefoe the LEF ethod eed a total coputatio with L * *log wi O N 4. Fo exaple, if L wi = 6, the local lat tack ad the local expoetial fae ethod eed O( 56* N ad O( 4* N coputatio epectively. The othe advatage of uig the local expoetial bealet i the availability of LCB coefficiet duig the popagatio, which ave oe coputatio tie fo local expoetial decopoitio. A we kow, the coputatio ode of D decopoitio i quaed, o the LEF ethod ave a lot of coputatio tie o agle decopoitio. I uay, the aalyi uig local expoetial fae i 3D cae i uch oe efficiet tha uig local lat tack ethod. Cocluio (e (f Figue 5. Acquiitio dip epoe (ADR ap uig LEF: < < 9, ϕ = 9 ; ADR fo dip agle ( ADR fo dip agle (, ϕ ( < < 9, ϕ = 9 = ; (e(fadr fo dip agle The let biefly dicu the coputatio efficiecy of the ethod. We copae thi ethod with the tie-couig local lat tack ethod. Fit, we coide D decopoitio ad aue the ae widow legth L wi fo Gauia widow ad bell widow, which coepod to the local lat tack ad local expoetial bealet decopoitio epectively. Fo local lat tack ethod, a lat tack coputatio with O( Lwi * L wi i equied fo each aple poit, which eult i a total coputatio with O( N* Lwi * L wi, whee N i the total ube of the aple poit. Howeve, fo local expoetial bealet decopoitio, we oly eed a LCT ad a LST We developed a efficiet ethod of 3D diectioal illuiatio aalyi i the local agle doai uig D local expoetial fae, which fo a tight-fae of edudacy 4. A the local coie/ie tafo have fat algoith, thi ethod ha uch bette efficiecy tha the local lat tack ethod. Nueical exaple of diectioal illuiatio (DI ap ad the acquiitio dip epoe (ADR fo the 3D SEG/EAGE alt odel illutated the validity ad efficiecy of the ew ethod. Fo the futhe wok, we will develop oe aplitude coectio ethod i local agle doai baed o the diectioal illuiatio aalyi to ipove the iage quality. Ackowledgeet We thak Xiaobi Xie, Ju Cao, Hui Yag, Jighuai Gao, Yaofeg He, Xiaofeg Jia fo helpful dicuio. The wok i uppoted by WTOPI (Wavelet Tafo O Popagatio ad Iagig fo eiic exploatio Poject. SEG Houto 9 Iteatioal Expoitio ad Aual Meetig 38

5 EDITED REFERENCES Note: Thi efeece lit i a copy-edited veio of the efeece lit ubitted by the autho. Refeece lit fo the 9 SEG Techical Poga Expaded Abtact have bee copy edited o that efeece povided with the olie etadata fo each pape will achieve a high degee of likig to cited ouce that appea o the Web. REFERENCES Auche, P., 994, Reak o the local Fouie bae, i J. J. Beedetto ad M. W. Fazie, ed., Wavelet, atheatica ad applicatio: CRC Pe, 3 8. Bea, G., C. Liu, R. Lu, D. Wille, ad I. Wato,, The cotuctio of ubuface illuiatio ad aplitude ap via ay tacig: The Leadig Edge, 9, Coifa, R. R., ad Y. Meye, 99, Reaque u l aalye de Fouie a feete: Copte Redu de l Acadeie de Sciece, Seie I 3, Luo, M. Q., J. Cao, X. B. Xie, ad R. S. Wu, 4, Copaio of illuiatio aalyi uig oe-way ad full-wave popagato: 7d Aual Iteatioal Meetig, SEG, Expaded Abtact, Mallat, S., 999, A wavelet tou of igal poceig, d ed.: Acadeic Pe Ic. Mao, J., ad Wu, R. S., 7, Illuiatio aalyi uig local expoetial bealet: 77th Aual Iteatioal Meetig, SEG, Expaded Abtact, Muedte, D., ad D. Ratcliff, a, Udetadig ubalt illuiatio though ay-tace odelig, Pat : Siple D alt odel: The Leadig Edge,, , c, Udetadig ubalt illuiatio though ay-tace odelig, Pat 3: Salt idge ad fuow, ad ipact of acquiitio oietatio: The Leadig Edge,, Muedte, D., M. Kelly, ad D. Ratcliff, b, Udetadig ubalt illuiatio though ay-tace odelig, Pat : Dippig alt bodie, alt peak, ad oecipocity of ubalt aplitude epoe: The Leadig Edge,, Scheide, W. A., ad G. A. Wibow, 999, Efficiet ad accuate odelig of 3-D eiic illuiatio: 69th Aual Iteatioal Meetig, SEG, Expaded Abtact, Wag, Y. Z., R. Cook, ad R. S. Wu, 3, 3D local coie bealet popagato: 73d Aual Iteatioal Meetig, SEG, Expaded Abtact, Wickehaue, M. V., 994, Adapted wavelet aalyi fo theoy to oftwae: A. K. Pete, Ltd. Wu, R. S., ad L. Che,, Mappig diectioal illuiatio ad acquiitio apetue efficacy by bealet popagato: 7d Aual Iteatioal Meetig, SEG, Expaded Abtact, , 3, Petack depth igatio i agle-doai uig bealet decopoitio: Local iage atix ad local AVA: 73d Aual Iteatioal Meetig, SEG, Expaded Abtact, , 6a, Diectioal illuiatio aalyi uig bealet decopoitio ad popagatio: Geophyic, 7, S47 S59., 6b, Taget-oieted bealet igatio baed o Gabo-Daubechie fae decopoitio: Geophyic, 7, S37 S5. Wu, R. S., L. Che, ad X. B. Xie, 3, Diectioal illuiatio ad acquiitio dip-epoe: 65th Aual Cofeece ad Exhibitio, EAGE, Exteded Abtact, P47. Wu, R. S., ad J. Mao, 7, Bealet igatio uig local haoic bae: 77th Aual Iteatioal Meetig, SEG, Expaded Abtact, Xie, X. B., S. W. Ji, ad R. S. Wu, 3, Thee-dieioal illuiatio aalyi uig wave-equatio baed popagato: 73d Aual Iteatioal Meetig, SEG, Expaded Abtact, , 6, Wave-equatio-baed eiic illuiatio aalyi: Geophyic, 7, S69 S77. Xie, X. B., ad R. S. Wu,, Extactig agle doai ifoatio fo igated wavefield: 7d Aual Iteatioal Meetig, SEG, Expaded Abtact, SEG Houto 9 Iteatioal Expoitio ad Aual Meetig 39

( ) ( x) SEG/San Antonio 2007 Annual Meeting. x x. x x. = is the local wavenumber. We will

( ) ( x) SEG/San Antonio 2007 Annual Meeting. x x. x x. = is the local wavenumber. We will Illuiatio aalyi ui local expoetial bealet Jia Mao* Ru-Sha Wu Modeli ad Iai aboatoy, IGPP, Uiveity of Califoia, Sata Cuz Viiti fo ititute of Wave ad Ifoatio, Xi a Jiaoto Uiveity, Xi a 70049, Chia Suay I