1. Super-wide angle beamlet propagator based on iterative wavefront reconstruction Zhongmou Xia, Ru-Shan Wu, Hong Liu 1 Modeling and Imaging Laboratory, IGPP, University of California, Santa Cruz Institute of Geology and Geophysics, Chinese Academy of Sciences

2.  Reviews of local cosine basis theory  Super-wide angle one-way scheme  Numerical tests  Conclusions 2 Outline

3. 1.Reviews of Local Cosine Basis Theory For 2D scalar acoustic equation: Corresponding one-way wave equation: Expression of analytic solution: Starting Point ！！！ (1 ) (2 ) (3 ) 3

4. (Wu et. al.,2008, Geophysics) 4

5. Local Cosine Basis The basis element The wavefield at depth Z can be decomposed into local cosine beamlets with windows along the horizontal x-axis coefficients of the decomposition beamlets (4) (5 ) (Coifman and Meryer, 1991) 5

6. (6) : propagator (For details see Wu et al., 2008), Wave equation in frequency and space domain: (7) Wavefield extrapolation expression: Old New 6

7. 7 B 2. Super-wide angle Scheme A c

8. Weight function (Jia and Wu, 2009, Geophysics,) : Cut angle 8

9. Weight function field in homogeneous medium 9

10. Two schemes to implement super-wide angle method Downward Horizongtal Weighted summation (Jia & Wu, 2009 ， Gepphysics) 10 Two schemes 1) “Interpolation method” 2) Iterative reconstruction method 1) “Interpolation method”

11. 2) Iterative reconstruction method 11 X0 + X Z0 + Z Z0 X0 - X X0 X0 + X Z0 + Z Z0 X0 - X X0 Combining Superposition Wavefield (5 points): X0 + XX0 - X X0 Z0 + Z Z0 The First Reconstructed Wavefront  To reconstruct the first wavefront

12. 12 Z0 + Z Z0 X0 - 2 X X0 Combining Superposition Wavefield( 9 points ): The Second Reconstructed Wavefront  To reconstruct the second wavefront Z0 + 2 Z X0 - 2 X X0 X0 - 2 X Z0 + Z Z0 Z0 + 2 Z Z0 + Z Z0 Z0 + 2 Z X0 - 2 X X0 X0 - 2 X

13. Z0 + Z Z0 Z0 + 2 Z X0 - 2 X X0 X0 + 2 X X0 - m X X0 + m X Z0 + m Z  All the reconstructed wavefronts 13

14. 3. Numerical tests Model1: 2D Salt Model (layered + salt model) (Made by Ruiyan) Model2: Bp Model 14 dx=dz=24m fd=15 HZ Nx=1001 Nz=150 dt=0.04s

15. Salt model at time 1.6s Snapshots for 2D Salt Model Regular one-way method Super-wide one-way LCB method RTM 15

16. Regular 16 Interpolation method Iterative reconstruction method

17. 12 3 4 13 9 17

18. Regular LCB Method Iterative Super-wide LCB Method

19. RTM MethodIterative Super-wide LCB Method

20. RTM MethodIterative Super-wide LCB Method Adding upward wavefields

21. D+H D+H+U

22. Conclusions 1. Super-wide angle beamlet propagator based on iterative wavefront reconstruction can handle large-angle and super- angle (e.g. turning waves) imaging problem. It can overcome the angle limitation but keep the merits of one-way method 2. Cost of super-wide angle beamlet (iterative reconstruction) one-way method is close 1.5-2 to regular one-way method. 22

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