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1 for strong-contrast meda and ts applcaton to subsalt mgraton Ru Yan 1* Ru-han Wu 1 Xao-B Xe 1 and Dave Walraven 1 GPP Earth and Planetary cences Department Unversty of Calforna anta Cru; Anadarko petroleum Corporaton ummary o overcome the weak scatterng lmtaton of tradtonal one-way elastc propagator we propose a hybrd elastc propagator for strong-contrast meda whch combnes the thn-slab propagator for weak perturbatons and wth the PO physcal optcs approxmaton at the rregular sharp boundares. he hybrd propagator shuttles between wavenumber and space doman: wave propagaton n wavenumber doman and heterogenetes nteracton n space doman renderng hgh effcency and accuracy of the propagator. Numercal test usng one smple salt model demonstrates the valdty of the hybrd propagator. Fnally the new propagator s appled to subsalt mgraton usng synthetc data generated based on D ubsalt model. ntroducton ubsalt exploraton can be very challengng due to the ssues posed by the often complex geometry of the salt bodes and the large mpedance contrasts assocated wth salt/sedment nterfaces. he large velocty contrast across the sedmentary/salt nterfaces together wth the frequently rugose character of these nterfaces can prevent sesmc waves from penetratng the salt body wth suffcent strength to mage subsalt structures. t s well known that converted-waves can penetrate a salt body and reach the blnd one of a P-wave survey. ome prevous work usng converted-wave magng Purnell 199; Kendall et al. 1998; Jones and Gaser 1999; Wu et al. 001 to mprove the P-wave llumnaton utled scalar wave propagators for both P- and -waves. t s noted that scalar propagator for elastc wave propagaton s dynamcally ncorrect. Prelmnary studes of converted wave magng usng elastc wave propagator have also been reported Wu et al Elastc thn-slab and elastc complex screen method Wu 1994; Wu and Xe 1994; Xe and Wu ; Wu and Wu 005; for a revew see Wu et al 007 are developed for one-way mgraton. t has ts specal feature and advantages n applyng to sesmc magng. he extrapolated wavefelds are vector wavefelds but P and modes are separated. hese specal features are especally useful for elastc wave magng n terms of controllng mgraton artfacts and parameter nverson. econdly oneway wave method s much more effcent often s orders of magntude faster than the full wave method. However those methods are based on perturbaton theory. hey can handle elastc perturbatons only up to 30% Wu and Wu 005 and may become nstable beyond ths lmt. Although these methods can be useful n reservor modelng and magng they fal to provde accurate wave propagaton n strong heterogeneous meda n whch salt or basalt exsts. o handle ths specfc case we propose to solve the boundary problem by applyng local reflecton/transmsson operator under the PO approxmaton or Krchhoff approxmaton and combne t wth thn-slab propagator for weak perturbatons n the framework of one-way marchng algorthm. n ths study we present the formulatons for the theory on hybrd elastc one-way propagator and descrbe ts three essental components n detal. Moreover we conduct a numercal test to verfy the accuracy and effcency of the hybrd elastc one-way propagator and also use the ubsalt model to demonstrate ts applcaton to sesmc magng. heory o handle the strongly heterogeneous meda wth sharp boundares such as salt or basalt nclusons we dvde the model nto dfferent domans along the sharp boundary Fgure 1. n each doman the wave feld can be computed wth the representaton ntegral: ux = Qx ' Gxx ; ' dω x' { ˆ ˆ } + [ n σx '] Gxx ; ' ux ' [ n Σxx ; '] d x' 1 x x' Ω ; = Ω where ux s the dsplacement feld at pont x wthn the G xx ; s Green s volume Ω enclosed by surface dsplacement tensor dyadc and Σ xx ; s Green s stress tensor tradc. ˆn s the surface normal as towards to the exteror of Ω. u x and σ x are the dsplacement and stress on the surface; Qx ' s body forces or equvalent body forces due to scatterng. he volume ntegral term yelds the contrbuton due to the sources nsde Ω whle the surface ntegral term that s Krchhoff ntegral accounts for the energy communcaton between dfferent domans. he tradtonal way to calculate the scattered wave n each doman s to solve the ntegral equaton along the boundary. However t nvolves huge computatons because the nteractons between all the boundary elements are consdered. Followng the sprt of the thn-slab propagator we solve the problem teratvely n a one-way fashon. he resultant operator s called hybrd elastc oneway propagator. Fgure 1 schematcally shows the realaton of a typcal hybrd propagator n strong heterogeneous medum wth EG Houston 013 Annual Meetng Page 3815

2 sharp boundary. As the velocty model s slced nto thn slab the sharp boundary s dscreted nto many boundary elements. he thn slab s separated nto two domans: hgh velocty one and low velocty one by two boundary elements. Wthn each doman the parameter varatons perturbatons are relatvely weak. hey are treated as volume scatterngs and handled by thn-slab propagator. Besdes the volume scatterngs each doman wll admt nternal reflectons and transmssons from adjacent domans at the boundary elements. hey together are called boundary scatterngs. We assume that the boundary elements wthn one thn slab are decoupled from each other so that the reverberatons between them are neglected. he dsplacement and tracton felds of the boundary scatterngs can be approxmately calculated by applyng reflecton/transmsson operator to the wave ncdent to the boundary element. t corresponds to the tangent plane approxmaton for smoothly curved boundary Voronorch When an ncdent wave enters a thn slab t wll nteract wth the volume heterogenetes as well as the boundary element n the current slab. he representaton ntegral wll gve the scattered wave whch wll be added to the ncdent wave at the ext of the thn slab. Based on the one-way propagaton prncple the wavefelds are updated teratvely step-by-step n the forward drecton wth no consderaton of the backward scatterngs. tradtonal thn-slab propagator. Dfferent domans have dfferent background veloctes so that the weak perturbatons assumptons of thn-slab propagator can be guaranteed n each doman. he nteracton wth the weak heterogenetes wll be calculated by the local Born approxmaton j = d d 0 V j u K xq x G0 K ; x x Ω Ω s doman and j 1 where Q0 x s the equvalent body force due to the nteracton of perturbatons and ncdent felds. For sotropc meda Q0 x = δρ x ω u0 x 3 + δλ x ε 0 x + δµ x ε 0 x where ρ s the densty; λ and µ are the Lame constants ε x are the s the unt tensor; u x and 0 0 space-doman dsplacement and stran feld of ncdent wave respectvely; P k 0 ; ˆ ˆ Kx + γ G K x = kke ρωγ 0 4 kβ 0 ; ˆ ˆ K x + γ G K x = kβkβ e 5 ρωγ 0 β are the background Green tensor of dsplacement n whch = γ k = K γ are the P and k K and β β wavenumber vectors respectvely wth K as the horontal wavenumber γ and γ β as vertcal wavenumbers k ˆ and k ˆ β as the unt vectors for P and waves respectvely. Fgure 1. chematc llustraton of the hybrd elastc one-way propagator. n the subsecton we wll descrbe three crtcal components of mgraton algorthms: thn-slab propagator for weak volume scatterngs background propagaton and reflecton/transmsson operator for sharp boundary scatterngs. 1. hn-slab propagator for weak volume scatterngs When several domans exst n a velocty model the wave propagaton n each doman wll be handled separately wth. Background propagatons he free propagatons background propagatons n the hybrd propagator are expressed as phase shfts n wavenumber doman: j 1 0 e γ β K x u Κ = d xe u0 x j 1 6 x j 1 Ω where Ω s doman. 3. ransmsson/reflecton operators on the boundary elements Boundary scatterngs are formulated n dfferent way from volume scatterngs n the thn slab. o compute the boundary scatterng we make a tangent plane approxmaton Voronovch 1999 whch assumes the boundary surface s smoothly curved so that the reflecton/transmsson coeffcents defned for an nfnte plane surface can be appled locally at each surface element. EG Houston 013 Annual Meetng Page 3816

3 We focus our attenton to one boundary element whch separates the medum nto upper and lower ones. Regardless of ther real doman numbers we set the ndex for the upper medum as 1 and the lower medum as. he ncdent waves of the upper and lower medum are gven U K and from the output of prevous thn slab as 01 U K n wavenumber doman. hese ncdent waves 0 nclude the scatterngs of the prevous boundary elements and the contrbutons from underneath are gnored. o accurately calculate the energy partton at the boundary element we transform the dsplacement of the ncdent waves at the upper and lower medums from the Cartesan coordnate to U 01 K and 0 K defned n the local boundary coordnate wth horontal axs parallel to the tangent of the local boundary. n the local boundary coordnate reflecton/transmsson coeffcents are appled to the ncdent wave to get the reflected and transmtted waves: P P PP P PP P 1 K R11 R K P P 1 K R11 R K P = PP P PP P P 7 K 1 1 R R 0 K P P 1 1 R R 0 U K U K where R and are reflecton and transmsson coeffcents calculated by Zoepprt s equaton Ak and Rchards he coeffcents are dependent on the horontal slowness n the local boundary coordnate. he frst and second subscrpts of the coeffcents are the medum ndexes. he superscrpts of the coeffcents specfy the wave types n the upper and lower medum respectvely. U = 1 s the dsplacement of the boundary scatterng whch s the sum of the nternal reflected wave and the transmtted wave from the other doman. / he tracton of the boundary scatterng P K / calculated from the dsplacement P K can be by the consttutve equaton n wavenumber doman. And then we transform both dsplacement and tracton nto space doman and back to the Cartesan coordnate. We pck the values at the boundary element and return to the doman they belongs to as U P / x and P / x. he scattered waves due to the boundary element at the slab ext can be calculated by Krchhoff ntegral n the horontal wavenumber doman Wu 1989 P / P / P / B j 0 j L P / P / 0 j; n d u K = x G K ; x 8 U x Γ K x ˆ where s the doman number; L s the boundary segment whch separates upper and lower medum. where 3 P k Γ 0 K ; x n ˆ = ρωγ 0 9 ˆ ˆ ˆ ˆ ˆ ˆ x + γ λ kn + µ nk kk K e 3 kβ Γ 0 K ; x n ˆ = ρωγ 0 β 10 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x + γ µ n kβ + kβn n kβ kβk K β e are the background Green s tractons. At the ext of each thn-slab the total feld s composed of three parts: the ncdent free-propagated wave u 0 the scattered feld by volume heterogenetes u V and the boundary scattered feld u B 1 K / x P u x j = d e 0 j 4π K u K 11 + u V K j + ub K j x j Ω where Ω s doman. Numercal tests Frst we test the elastc one-way propagator usng a salt wth a wedge shape embedded n a homogeneous meda. Fgure llustrates the velocty model wth elastc parameters. he source s a 15 H Rcker wavelet and located n the center of the model. n ths smple case the PO approxmaton s expected to work the best. llustrated n the upper panel of Fgure 3 are the horontal and vertcal dsplacements calculated by elastc one-way propagator. Compared wth the snapshots calculated from full-wave fnte dfference FD the lower panel of Fgure 3 the transmtted wavefront of the elastc one-way propagator matches wth them very well. he energy partton at the nterface s almost the same as full-wave FD. Next we move on to a more complex model D velocty profle smulatng the ubsalt model Fgure 4. On top of the model s a water layer. FD cannot solve the flud-sold problem very well. On the contrary the hybrd propagator can do a good job to handle the nterface. he model s dvded nto three dfferent domans: water sedment and salt. Each doman has ts own background and perturbaton parameters. We desgn an observaton system and conduct EG Houston 013 Annual Meetng Page 3817

4 a sesmc experment. he acquston system was comprsed of 301 shots from 7000 m to m wth an nterval of 100 m. Each shot was recorded by a lne of recevers wth double-spread confguraton. he number of recevers was 561 and the maxmal offset s 7000 m. Both sources and recevers are located on the water surface. he recorded data are pressure only smulatng a hydrophone response. he source s a 15 H Rcker wavelet and the total recordng tme s 1 s wth a tme nterval of 0.01 s. he synthetc data were modeled by the esseral D applcaton package usng a FD approach. We mgrate the sesmc data wth the hybrd one-way propagator and obtan the PP P P and mages. Here we only show PP mage n Fgure 5 because t has the hghest sgnal-to-nose rato. From the mgraton mage a large porton of subsalt reflectors can be clearly dentfed except the secton wth very large dps. here are some mgraton artfacts near the true mages due to multples and cross-talk. Preprocessng the sesmc data wll help to mprove the qualty of the mgraton mage. We developed the theory and method of a hybrd elastc one-way operator whch combnes the elastc thn-slab propagator n weakly heterogeneous meda and reflecton/transmsson operators at the sharp boundares. he accuracy and effcency of the dual-doman depthmarchng propagator are valdated by a smple numercal example. he applcaton of the propagator to sesmc magng s demonstrated on ubsalt model and three subsalt reflectors are shown clearly n the resulted mgraton mage. Fgure. An elastc salt model. he elastc parameters and shot locaton are ndcated n the fgure. Fgure 4. he D ubsalt model: a P-wave velocty; b -wave velocty and c densty. Fgure 3. he snapshots generated n the wedge model by the hybrd elastc propagator upper panel and fnte dfference lower panel. he horontal components are shown on the left and the vertcal components are shown on the rght. Conclusons Acknowledgements he research work s funded by DEEPAR project. he authors acknowledge the support from WOP consortum at UC anta Cru. he authors thank Anadarko for provdng the D ubsalt model and the help from the techncans n etrale Company n generatng the synthetc data for the model. he authors apprecate the helpful dscussons wth Zengx Ge and Yngca Zheng. Fgure 5. he PP mage of D ubsalt model generated by elastc one-way propagator. EG Houston 013 Annual Meetng Page 3818

5 EDED REFERENCE Note: hs reference lst s a copy-edted verson of the reference lst submtted by the author. Reference lsts for the 013 EG echncal Program Expanded Abstracts have been copy edted so that references provded wth the onlne metadata for each paper wll acheve a hgh degree of lnkng to cted sources that appear on the Web. REFERENCE Ak K. and P. G. Rchards 1980 Quanttatve sesmology: heory and methods W. H. Freeman and Co. Jones N. and J. Gaser 1999 magng beneath hgh-velocty layers: 68 th Annual nternatonal Meetng EG Expanded Abstracts Kendall R. R.. H. Gray and G. E. Murphy 1998 ubsalt magng usng prestack depth mgraton of converted waves: Mahogany Feld Gulf of Mexco: Annual nternatonal Meetng EG Expanded Abstracts Purnell G. W. 199 magng beneath a hgh-velocty layer usng converted wave: Geophyscs Voronovch A. G Wave scatterng from rough surface: prnger-verlag. Wu R Representaton ntegrals for elastc wave propagaton contanng ether the dsplacement term or the stress term alone : Physcal Revew Letters 6 no Wu R Wde-angle elastc wave one-way propagaton n heterogeneous meda and an elastc wave complex-screen method: Journal of Geophyscal Research 99 B Wu R.. H. Guan and X. Y. Wu 001 magng steep subsalt structure usng converted wave paths: Presented at the 71 st Annual nternatonal Meetng EG. Wu R.. and X. B. Xe 1994 Multscreen back propagator for fast 3D elastc prestack mgraton: Mathematcal Methods n Geophyscal magng PE Wu R.. X. B. Xe and X. Y. Wu 007 One-way and one-return approxmatons for fast elastc wave modelng n complex meda n R.. Wu and V. Maupn eds. Advances n wave propagaton n heterogeneous earth: Elsever Wu R.. R. Yan X. B. Xe and D. Walraven 010 Elastc converted-wave path mgraton for subsalt magng: 80 th Annual nternatonal Meetng EG Expanded Abstracts Wu X. Y. and R.. Wu 006 AVO modelng usng elastc thn-slab method: Geophyscs 71 no. 5 C57 C67 Xe X. B. and R.. Wu 1995 A complex-screen method for modelng elastc wave reflectons: 65 th Annual nternatonal Meetng EG Expanded Abstracts Xe X. B. and R.. Wu 001 Modelng elastc wave forward propagaton and reflecton usng the complex screen method: he Journal of the Acoustcal ocety of Amerca 109 no Xe X. B. and R.. Wu 005 Multcomponent prestack depth mgraton us ng elastc screen method: Geophyscs 70 no EG Houston 013 Annual Meetng Page 3819