2011 SEG SEG San Antonio 2011 Annual Meeting 3377

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1 Source-receiver pretack depth migration uing dreamlet Bangyu Wu *, Ru-han Wu and Jinghuai Gao Modeling and Imaging Laboratory, IGPP, Univerity of California, Santa Cruz Summary The dreamlet method add the time-frequency localization to the frequency domain beamlet method by an additional wavelet tranform along the time axi. Thi method ha been applied to hot-profile pretack depth migration and how great potential on wavefield compreion and good imaging quality. However, in hot domain migration, the point ource excited wavefield expand during migration on the time-pace plane and then increae the amount of dreamlet coefficient and therefore, the computation time. The urvey inking migration not only downward continue the geophone but alo the hot. Combining the urvey inking and dreamlet decompoition can achieve higher compreion ratio on the whole data volume. For the SEG/EAGE model, the dreamlet urvey inking algorithm i time fater compared with the hot domain dreamlet method with imilar imaging quality. Introduction Reflection eimic data i collected in hot gather. Each gather i a point-ource excited wavefield recorded by a receiver array on the urface. Two common categorie of wave-equation migration are available in the indutry: Survey Sinking Migration (SSM) and hot domain migration (SDM). The SDM imaging proce i to downward extrapolate the point ource field and the received reflection field independently, and then apply the imaging condition at imaging point. For offhore data acquiition, each hot ha a limited receiver aperture, but the hot number can be extremely large. In addition, the downward extrapolation of the ource field need a rather large aperture to include all the wave path. Therefore hotdomain pretack depth migration i time-conuming and inefficient. SSM method are better uited for marine towed cable acquiition geometrie than SDM and can be order of magnitude fater in runtime (Dimitri Bevc etal., ). Dreamlet (drumbeat-beamlet) (Wu etal., 8;Wu etal., 9, ) i a type of wavelet bai for ignal decompoition along both patial and time axe, where drumbeat i the time-frequency atom and beamlet i pacewavenumber atom. The dreamlet atom are localized in both time and pace and contitute a complete et of frame or bai to repreent wavefield. The dreamlet decompoition lead to a very efficient repreentation of eimic data, and often large data compreion ratio can be achieved. The dreamlet propagator i formulated with a local-background-velocity propagator for large-cale tructure and a local phae-creen correction for mall cale perturbation. So the pretack migration can tay in the compreed domain with high wavefield compreion ratio. Beamlet urvey inking pretack depth migration wa invetigated in offet domain (Luo etal., 6). In thi paper, we propoe a dreamlet urvey inking cheme for pretack depth migration. The Local Coine Bai (LCB) Tranform i applied to the common ource, common beamlet receiver and time axe, which i a full wavefield decompoition uing dreamlet atom. Another advantage of the dreamlet urvey inking i that the inking ytem keep only the ueful coefficient for imaging the tructure beneath the ytem at each depth. Thi property i a major factor to peed up the migration proce. A three-catter model i ued to demontrate thi property. Uing the SEG/EAGE model, we tet the imaging quality of the dreamlet urvey-inking pretack migration, and invetigate the computational apect on complex geologie. The hot domain dreamlet migration reult i hown for comparion. Dreamlet decompoition of pretack eimic data In thi application, we ue the orthogonal dreamlet for the wavefield decompoition, that i, to ue a threedimenional LCB decompoition to tranform the dataet from point ource, point receiver and time record into a beamlet-ource, beamlet-receiver and drumbeat (timefrequency localized atom) record. LCB ue overlapped bell function to window the coine ocillation and i developed a a type of orthogonal wavelet bai (Mallat, 998). In the pace domain, the local coine bai element can be characterized by poition x, the interval (the n nominal length of the window) Ln = xn+ xn, and wavenumber index m ( m =,, M-, M denote the total ample point of the interval) a follow bmn ( x) = Bn ( x) co( ξm ( x xn )) () L n Where π ξm = m + and Bn ( x ) i a bell function which Ln i mooth and upported in the compact interval [ xn ε, xn+ + ε '] for xn ε xn+ + ε ', where ε, ε ' a the left and right overlapping radiu. Viiting from Intitute of Wave and Information, Xi an Jiaotong Univerity, Xi an, China Intitute of Wave and Information, Xi an Jiaotong Univerity, Xi an, China SEG SEG San Antonio Annual Meeting 77

2 Source-receiver pretack depth migration uing Dreamlet () = () co( ω ( )) b t B t t t ij j i j Tj π and ωi = i +. Figure i a chematic illutration Tj about reorting the hot-gather eimogram and the computing aperture. Receiver location ( Km ) Source location ( Km ) Common receiver Reiceived eimic data Common ource Zero padding Zero padding Figure : Schematic illutration of the dreamlet urvey inking wavefield reorting for the SEG/EAGE alt model acquiition ytem. The time axi i perpendicular to the ource-receiver plane. And the blue part i zero padding for the calculation aperture. The trip in the middle of the plane i the original eimic data. For each common ource gather, we decompoe the geophone array data into receiver dreamlet domain U x,; tx Ut (, ω, x, ξ ; x) = U x,; tx, d( xt,) () ( g ) g g ( g ) Where d tand for dreamlet atom and μ μ = ( x, ξ, t, ω) i the local parameter et. Then for each common dreamlet receiver, decompoe along the ource line into beamlet ource U( t, ω, xg, ξ g; x ) U( t, ω, xg, ξ g; x, ξ ) () = U t, ω, x, ξ ; x, b ( x) ( ) g g mn Survey inking the wavefield in dreamlet domain The dreamlet urvey inking method involve the following tep: (a) Dreamlet decompoition of the wavefield U t, x, x to get the wavefield dreamlet ( g ) coefficient U( t,, xg, g, x, ) ω ξ ξ. μ () (b) For each depth, firt downward extrapolate the receiver wavefield uing dreamlet propagator x, ξ ( ', ω', ' g, ξ' g; +Δ ) = G ( t', ω ', x' g, ξ' g; t, ω, xg, ξ g) U, ( t, ω, xg, ξ x ) ξ g U t x z z t, ω, xg, ξg (c) Then ink the ource for the already downward continued wavefield uing the dreamlet propagator again U ', ' ( t'', ω'', x'', ξ'' ; z+δ z) = x g ξ g (6) G t'', ω '', x'', ξ'' ; t', ω', x', ξ' U t', ω', x', ξ' t', ω', x', ξ' ( ) x', ' ( ) g ξ g (5) (d) Tranform the wavefield to the pace-frequency domain for the local phae creen correction. (e) Apply the imaging condition at each depth, extract the wavefield amplitude at zero time zero offet a the image amplitude. We detail the local phae creen correction algorithm a follow: (a) For every frequency multiply the wavefield U x, x, ω with the phae-creen, ( g ) exp iω( + ) Δz. V( x) V( xg) Where V( x ) and V( xg ) correpond to the ource and receiver location velocitie, repectively. (b) Fold the wavefield uing the bell function along the common ource and common receiver axe (a tep of LCB tranform). (c) Subtract the background phae-creen from the phae-creen correction in (a) uing the local background velocitie, exp iω + Δz. V( xn) V( xgn) (d) Unfold the wavefield (a tep of invere LCB tranform). Numerical example We ue a three-catter model to examine the diffraction hyperbola and it evolution after migration by the deramlet urvey inking method (Figure and ). Figure : Dreamlet urvey inking migration reult for the three catter model. SEG SEG San Antonio Annual Meeting 78

3 Source-receiver pretack depth migration uing Dreamlet (a) Ditance (km) (c) (e) (b) Ditance (km) (d) (f) where the firt and third catter are located. For comparion, we alo put the eimogram at the ame ource location from the hot domain dreamlet migration in the ame depth, hown in Figure (b, d, f, h). In the SDM, the image at each depth i produced by correlating ource and receiver wavefield. From thee figure, we ee that the dreamlet propagator alway move the eimogram toward zero time and doen t wrap around. So the time axi will be horter a the depth increaing. The information ued to image the upper tructure i not been migrated anymore. At every imaging point, the urvey inking will abandon all the ued data automatically. Thi i one apect in which the dreamlet urvey inking will peed up the migration computation. Next, the SEG/EAGE A-A model i ued to demontrate thi method on complex geological tructure. The acquiition ha 5 hot with left-hand-ide receiver and the maximum number of receiver for one hot i 76. The original velocity model ha ample in the horizontal extend and 5 ample in depth, both the ample interval are 8 feet. (a) (b) (g) (h) Figure : (a) Dreamlet urvey inking migration reult; (b) i the image by hot domain dreamlet method. Figure : The common ource eimogram from urvey inking (on the left)) and from hot domain (on the right). (a, b) are from the firt depth and (g, h) from the lat depth. (c, d) and (e, f) are from the depth where the firt and third catter located. Common ource eimogram are extracted in different depth for the dreamlet urvey inking, hown a Figure (a, c, e, g). Figure (a) and (g) are eimogram at the firt and lat depth; (c, e) are the eimogram at the depth Figure (a) how the dreamlet urvey inking depth image. Figure (b) i the dreamlet SDM reult for comparion. Generally peaking, Figure (a, b) have imilar imaging quality. But the dreamlet SSM i at leat three time fater than the SDM dreamlet method on thi model. The 75th hot receiver ide eimogram from SDM dreamlet pretack depth migration i hown in Figure 5( b, d, f, h) and the correponding common ource from the SSM dreamlet are in Figure 5 (a, c, e, g). We ue the minimum and maximum velocity in the model to evaluate the SEG SEG San Antonio Annual Meeting 79

4 Source-receiver pretack depth migration uing Dreamlet traveltime and cut the eimogram in SDM method for computation efficiency. So there are truncation in the eimogram. (a) Ditance (km) (b) Ditance (km) (c) (d) SEG/EAGE model. The black line tand for the urvey inking dreamlet coefficient. The red line and the blue line are ource and receiver wavefield dreamlet coefficient in the hot domain repectively, and the purple line, their um. We ee that at the beginning of the migration, the urvey inking dreamlet coefficient have a larger amount than the hot domain. Thi i becaue of the interpolation during the ource inking proce in the hallow depth. After propagation, the ource wavefield will pread and produce more equivalent ource. That mean the common ource or receiver gather will increae after propagation. However, after inking to greater depth, the dreamlet coefficient are dratically decreaed in the SSM cheme. Thi i why the SSM i much more efficient than the hot domain migration. (e) (f) (g) (h) Figure 6 Variation of dreamlet coefficient amount during migration. The black line i for the urvey inking dreamlet coefficient; the red line and the blue line are the ource and receiver dreamlet coefficient in the hot domain repectively, and the purple line, their um. Concluion We applied the urvey-inking cheme to the dreamlet pretack depth migration. The reult demontrate the high quality imaging of thi method and good computational efficiency. The dreamlet propagator doe not wrap around the eimic data on the time axi and the ued eimic data i abandoned automatically during the migration proce. The combination of dreamlet decompoition and urveyinking cheme in migration can further increae the data compreion ratio and ha the potential to image in the compreed domain. Figure 5: The common ource eimogram from SSM (a, c, e, g) and the ame common ource from SDM (b, d, f, h) for depth (a, b), 5 (c, d), (e, f) and 5 (g, h). Figure 6 how the comparion of dreamlet coefficient amount between the SSM and SDM at every depth for the Acknowledgment The author thank Yueming Ye, Haoran Ren at UCSC and Charle C. Moher and Jun Cao from ConocoPhillip for helpful dicuion on the concept of Survey Sinking, And thank Rui Yan for her help generating the three-catter model. Thi work i upported by WTOPI (Wavelet Tranform On Propagation and Imaging for eimic exploration) Project at Univerity of California, Santa Cruz. SEG SEG San Antonio Annual Meeting 8

5 EDITED REFERENCES Note: Thi reference lit i a copy-edited verion of the reference lit ubmitted by the author. Reference lit for the SEG Technical Program Expanded Abtract have been copy edited o that reference provided with the online metadata for each paper will achieve a high degree of linking to cited ource that appear on the Web. REFERENCES Bevc, D., M. Fliedner, S. Crawley, and B. Biondi,, Wave equation imaging comparion: Survey inking v. hot profile method: 7rd Annual International Meeting, SEG, Expanded Abtract, Claerbout, J. F., 985, Imaging the earth interior: Blackwell. Jin, S., C. C. Moher, and R. S. Wu,, Offet-domain peudocreen pretack depth migration: Geophyic, 67, 895 9, doi:.9/ Luo, M., and S. Jin, 6, Offet-domain LCB beamlet pretack depth migration: 76th Annual International Meeting, SEG, Expanded Abtract, Mallat, S., 999, A wavelet tour of ignal proceing, nd ed.: Academic Pre. Wang, Y. Z., and R. S. Wu,, Seimic data compreion by adaptive local coine/ine tranform: Geophyical Propecting, 8, 9, doi:.6/j x. Wu, B., R. S. Wu, and J. Gao, 9, Dreamlet pretack depth migration uing local coine bai and local exponential frame: 79th Annual International Meeting, SEG, Expanded Abtract, Wu, B., R. S. Wu, and J. Gao,, Orthogonal dreamlet decompoition and it application to eimic imaging: 8th Annual International Meeting, SEG, Expanded Abtract, Wu, R. S., B. Y. Wu, and Y. Geng, 8, Seimic wave propagation and imaging uing time-pace wavelet: 78th Annual International Meeting, SEG, Expanded Abtract, Wu, R. S., B. Wu, and Y. Geng, 9, Imaging in compreed domain uing dreamlet: International Geophyical Conference, CPS/SEG, Expanded Abtract, 57. Wu, R. S., L. Chen, and Y. Wang,, Synthetic beam-ource and plane-ource for pretack beamlet migration: 7nd Annual International Meeting, SEG, Expanded Abtract, 6 9. SEG SEG San Antonio Annual Meeting 8