( ) ( x ) is a bell function which is smooth and. Visiting from institute of Wave and Information, Xi an Jiaotong University, Xi an , China

Transcription

1 3D illuination analyi uing local exponential fae Jian Mao* Ru-Shan Wu Modeling and Iaging Laboatoy IGPP Univeity of Califonia Santa Cuz Viiting fo intitute of Wave and Infoation Xi an Jiaotong Univeity Xi an 749 China Suay In thi pape we develop an efficient ethod of diectional illuination analyi fo 3D cae uing local exponential fae. To get the local angle infoation the wave field i decopoed by D local exponential fae which fo a tight fae of edundancy 4 ipleented by the cobination of local coine and ine tanfo. A the local coine/ine tanfo have fat algoith thi ethod povide an efficient tool of illuination analyi which i uch fate than foally ued local lant tack ethod. We calculated the diectional illuination (DI) ap and the acquiition dip epone (ADR) fo the 3D SEG/EAGE alt odel to deontate the validity of ou ethod. Intoduction Seiic illuination analyi i a ueful tool that give potential detecting powe of a pecific acquiition yte fo a given tuctue. Taditionally illuination analyi i baed on the ay tacing technique (Schneide 999; Bea et al. ; Muedte et al. abc). The ay-baed ethod can povide both the intenity and diection infoation caied in the wavefield. Howeve the high fequency ayptotic appoxiation and the ingulaity poble of the ay theoy ay eveely liit it accuacy in coplex egion (Hoffann ). Full-wave finitediffeence ethod i widely ued fo wave popagation iulation. But it uually povide only the total illuination and it too expenive fo pactical application fo illuination analyi. Recently technique to obtain the localized angle doain infoation fo the fequency doain wave field baed on bealet decopoition o local lant tack have been developed and applied to diectional illuination analyi (Wu and Chen 3 6; Xie and Wu ; Xie et al. 6). The Gabo-Daubechie fae decopoition i coplete but not othogonal and theefoe ha edundancy in the epeentation. Local lant tack fo wavefield decopoition ae coputation deanding. The local coine bai (LCB) decopoition on the othe hand i othogonal. Fat algoith of local coine/ine tanfo exit. Howeve local coine bealet have alway two lobe in yety with epect to the vetical axi. Thi lack of uniquely defined diection-localization pevent it ue fo diectional illuination analyi. To ovecoe thi hotcoing a local exponential fae (LEF) ethod (Mao and Wu 7) have been uccefully developed fo D diectional illuination analyi with high efficiency. In thi pape we extend thi ethod to the 3D cae. Fit we intoduce the D local exponential fae (LEF) and it application to wavefield decopoition in 3D cae and then give a bief deciption on how to calculate the diectional enegy fluxe fo the wavefield and apply the to the illuination analyi. To deontate the potential application of thi ethod illuination exaple ae calculated fo the 3D SEG/EAGE Salt odel. D Local coine/ine bae and local exponential fae The Balian-Low theoe indicate that othogonal bae with good fequency localization geneated by Gau windowed exponential function do not exit. Nevethele local coine/ine bae wa uccefully contucted by Coiffan and Meye (99) (ee alo Mallat 999) which conit of coine/ine ultiplied by ooth copactly uppoted bell function. In fact thee othonoal bae ae foed by linea cobination fo a tight fae of local exponential (Daubechie et al. 99; Auche 994). In thi pape we will popagate the wavefield uing othonoal bae fo the popagation efficiency. Then we can go back to the tight fae epeentation with local exponential fo coplete local angle-doain infoation. The local haonic (coine/ine) bai eleent can be chaacteized by poition x n the noinal length of window L = x x and wavenube index and n n+ n ( c) x xn ψn = Bn co π + () x xn ψn = Bn in π + () whee Bn ( x ) i a bell function which i ooth and uppoted in the copact inteval [ xn ε xn + ε ] fo xn + ε xn+ ε with ε ε a the left and ight ovelapping adiu epectively. We define + φ = b x + ib x = B x exp i x x ξ ( + ) ( c) n n n n n φ = b x ib x = B x exp i x x ξ ( ) ( c) n n n n n (3)

2 3D illuination analyi uing local exponential fae π + whee ξ = i the local wavenube. We will ( call φ + ) ( n and φ ) n a ight- and left-popagating local exponential bealet. Then we can eaily get the D local haonic ( cc ) b b c b c and b. Take b c a exaple b c ( ) c x y ψn x ψ pq ( y) =. (4) So do the D local exponential fae g ++ g + g + and. Take a exaple g g + ( + ) φ + n φ pq g ( xy ) = ( y). (5) And the edundancy of the fae i 4 fo D decopoition. The local exponential fae ae linea cobination of local haonic fo exaple ( ++ ) ( + ) g ( x y) + g ( x y) cc b ( x y) =. (6) ( + ) ( ) 4 + g ( xy ) + g ( xy ) Fo the local plane wave of local hoizontal wavenube ξ = ξ ξ the coeponding popagating angle i ( p) ξ + ξ p ξ θ = ( θ ) acin actan ϕp = k ξ p (9) Whee θ and ϕ p ae the local incident angle with epect to the z-axi and x-axi epectively (hown in Figue ). θ i fo to 9 and ϕ p i fo 8 to 8. k = ω v the velocity at (xz) whee x=(xy). i the local wavenube and v ( x z) ( z) x i Wave field decopoition in local wavenube doain The fequency doain wave field uxyzω ( ) at depth z can be decopoed into local exponential bealet with window along the x-axi and y-axi ( ++ ) ( ++ ) ( xn yq ξ ξp g ( x y) ( + ) ( + ) + ( xn yq ξ ξp g( x y) uxyz ( = ( + ) ( + ) n q p + ( xn yq ξ ξp g ( x y) ( ) ( ) uˆ + z ( xn yq ξ ξp g( x y) (7) ( ++ whee ) ( + g ) ( + g ) ( g and g ) ae the local exponential decopoition fae. ( ++ ) ( + ) uˆ ( x y ξ ξ uˆ ( x y ξ ξ z n q p ( + ) ( xn yq ξ ξp and z n q p ( ) ( xn yq ξ ξp the coefficient of the local exponential fae decopoition epectively located at pace window ( xn y q) and wavenube window ( ξ ξ p). The coefficient coeponding to local exponential bealet can be calculated by local coine/ine tanfo ( ++ ) take uˆ ( z) a exaple ( ++ ) ( cc) ( c) ( c) uˆ ( ˆ ˆ ˆ ˆ z = u z iu z iu z u ( z) )(8) 6 ( cc) ( c) ( c) whee uˆ ( z ) uˆ ( z ) uˆ ( z ) and uˆ ( z) ae the coplex coefficient of local haonic decopoition of the wave field. The othe coefficient can be calculated iilaly. ae Figue. Definition of of incident angle = ( θ ϕ ) cae Illuination analyi in the local angle doain θ fo 3D Fo a given acquiition geoety the fequency-pace Geen function fo ouce to ubuface point (xz) can be decopoed at the iage egion to a uation of local wavenube coponent. That i G( x z = G( x z θ () θ Whee G( z function and G( z x i the fequency-pace Geen i it local-anglecoponent at θ. Siilaly the fequency-pace Geen function fo ubuface point (xz) to eceive can be decopoed G( x z = G( x z θ () θ Whee G( z function and G( z x i the fequency-pace Geen i it local-anglecoponent at θ.

3 3D illuination analyi uing local exponential fae In ode to evaluate the local angle doain illuination enegy ditibution fo a given acquiition yte in a velocity odel we u up the contibution fo all the ouce at the iage point fo each local angle. That i D ( ) ( ) a x z θ ω = G x z θ ω () whee D ( z a i the diectional illuination ap. Then we can get the total illuination by the uation of all the angle D x z ω G x z θ ω (3) total = θ In ode to evaluate the apetue and popagation effect of the given acquiition geoety on enegy ditibution fo a pecific pai of incident/eceiving angle we ue unit ipule a ouce at both ouce and eceive point fo the entie acquiition configuation and aue a unit catteing coefficient at each pace point. Siila to the pocedue of DI apping we u up contibution of the Geen function fo each incident/eceiving pai to get the acquiition apetue efficacy (AAE) atix at each iage point which neglect the detailed wave inteface patten and conide only the enegy ditibution in pace and angle of the acquiition configuation. Then the AAE atix at point (xz) i defined a E x z θ θ ω ( g ) = G( x z θ ) ( ) ω G x z θg ω (4) We can futhe educe the acquiition efficacy atix at each point to a function of eflecto dip by uing up all the eflected enegy fo each eflecto. Ad ( x z θn = E( x z θn θ (5) Whee A ( z θ d n i the ADR (acquiition dip epone) vecto fo point (xz) and θ n i the eflecto noal angle (igation-dip angle) to the vetical and θ i the eflection angle with epect to the noal which can be obtained fo θ and θ g. The value of the ADR ap eaue the dip-angle epone of the acquiition yte including the ouce and eceive apetue and popagation effect. Nueical exaple To deontate the application of the illuination analyi we calculated a nube of nueical exaple uing the 3D SEG/EAGE alt velocity odel. Figue. location of the 45 hot location of the lice fo copaion Figue 3. Vetical and hoizontal lice of velocity odel Thi exaple iulate the illuination condition of the 45 hot data et. The data et epeent a land type acquiition geoety. The location of thoe 45 hot ae hown in the Figue. Figue 3 how the vetical and hoizontal lice of the velocity odel and the location of the lice ae aked in Figue. Figue 4 i the eult of DI ap of diffeent incident angle θ which ae ( θ = 3 ϕ = 9 ) and ( θ = 3 ϕ = 9 ) epectively. Fo the DI ap we can ee the enegy flux of diffeent local angle clealy epecially fo the ubalt aea. Fo ADR calculation to get each eceive Geen function need a lage aount of coputation and then we jut ue oe apled eceive Geen function to get the ADR ap. Figue 5 how the ADR ap calculated by the local exponential fae ethod the dip angle ae ( < θ < 9 ϕ = 9 ) and ( < θ < 9 ϕ = 9 ) which epeent negative and poitive dip epone epectively. Thee figue how the diffeent epone fo diffeent dip angle which i due to the alt tuctue and the acquiition yte. Fo figue 4 we ee that the

4 3D illuination analyi uing local exponential fae illuination with negative aziuthal angle i uch pooe than that of poitive aziuthal angle. Coepondingly in figue 5 the dip epone fo eflecto with negative dip ae weake than that with poitive dip. Fo thee illuination eult we can alo explain the poo iage quality of the ubalt aea fo eflecto with cetain dip angle. (e) (c) (f) Figue 5. Acquiition dip epone (ADR) ap uing LEF: < θ < 9 ϕ = 9 ; (c)(d) ADR fo dip angle ADR fo dip angle ( < θ < 9 ϕ = 9 ) Concluion (d) Figue 4. Diectional illuination (DI) ap uing LEF: θ = 3 ϕ = 9 ; (c)(d) DI fo DI fo angle ( ) angle ( θ = 3 ϕ = 9 ) We developed an efficient ethod of 3D diectional illuination analyi in the local angle doain uing D local exponential bealet which fo a tight-fae of edundancy 4. A the local coine/ine tanfo have fat algoith thi ethod ha uch bette efficiency than the local lant tack ethod. Nueical exaple of diectional illuination (DI) ap and the acquiition dip epone (ADR) fo the 3D SEG/EAGE alt odel illutated the validity and efficiency of the new ethod. Fo the futhe wok we ay develop oe aplitude coection ethod in local angle doain to ipove the iage quality. Acknowledgeent We thank Xiaobi Xie Jun Cao Jinghuai Gao Yaofeng He Xiaofeng Jia fo helpful dicuion and Hui Yang fo poga help. The wok i uppoted by WTOPI (Wavelet Tanfo On Popagation and Iaging fo eiic exploation) Poject and the DOE/Baic Enegy Science poject at Univeity of Califonia Santa Cuz.

5 3D illuination analyi uing local exponential fae Refeence Auche P. 994 Reak on the local Fouie bae in Wavelet Matheatica and application edited by J.J. Benedetto and M.W. Fazie: CRC Pe 3-8. Bea G. C. Liu R. Lu D.Willen and I. Waton The contuction of ubuface illuination and aplitude ap via ay tacing: The Leading Edge CoifanR.R. and Meye Y. 99 Reaque u l analye de Fouie a fenete Copte Rendu de l Acadeie de Science Pai Seie I Luo M.Q. J. Cao X.B. Xie and R.S. Wu 4 Copaion of illuination analyi uing one-way and full-wave popagato: 7nd Annual Intenational Meeting SEG Expanded abtact SEG Mallat S. 999 A Wavelet Tou of Signal Poceing econd edition Acadeic Pe. Mao J. and Wu R.S. 7 Illuination analyi uing local exponential bealet: 77th Annual Intenational Meeting SEG Expanded Abtact Muedte D. and D. Ratcliff a Undetanding ubalt illuination though ay-tace odeling Pat : Siple D alt odel: The Leading Edge Muedte D. M. Kelly and D. Ratcliff b Undetanding ubalt illuination though ay-tace odeling Pat : Dipping alt bodie alt peak and nonecipocity of ubalt aplitude epone: The Leading Edge Muedte D. and D. Ratcliff c Undetanding ubalt illuination though ay-tace odeling Pat 3: Salt idge and fuow and ipact of acquiition oientation: The Leading Edge SchneideW. A. and G. A.Winbow 999 Efficient and accuate odeling of 3-D eiic illuination: 69th Annual Intenational Meeting SEG Expanded Abtact Wang Y.Z. R. Cook and R.S. Wu 3 3D local coine bealet popagato: 73d Annual Intenational Meeting SEG Expanded Abtact Wickehaue M.V. 994 Adapted wavelet analyi fo theoy to oftwae: A K Pete. Wu R. S. and L. Chen Mapping diectional illuination and acquiition apetue efficacy by bealet popagato: 7nd Annual Intenational Meeting SEG Expanded Abtact Petack depth igation in angle-doain uing bealet decopoition: Local iage atix and local AVA: 73dAnnual Intenational Meeting SEG Expanded Abtact a Diectional illuination analyi uing bealet decopoition and popagationgeophyic 7 S47-S59. 6b Taget-oiented bealet igation baed on Gabo-Daubechie fae decopoition: Geophyic 7 S37 S5. Wu R. S. L. Chen and X. B. Xie 3 Diectional illuination and acquiition dip-epone 65th Annual Confeence and Exhibition EAGE Extended Abtact P47. Wu R.S. and J. Mao 7 Bealet igation uing local haonic bae: 77th Annual Intenational Meeting SEG Expanded Abtact Xie X. B. and R. S. Wu Extacting angle doain infoation fo igated wavefield: 7nd Annual Intenational Meeting SEG Expanded Abtact Xie X. B. S.W. Jin and R. S. Wu 3 Theedienional illuination analyi uing wave-equation baed popagato: 73d Annual Intenational Meeting SEG Expanded Abtact Wave-equation-baed eiic illuination analyi Geophyic 7 S69-S77.