Super-wide angle beamlet propagator based on iterative wavefront reconstruction

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

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

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

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

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

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

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

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 1) “Interpolation method” Two schemes 2) Iterative reconstruction method 1) “Interpolation method” (Jia & Wu, 2009，Gepphysics) Downward Horizongtal Weighted summation 10

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

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

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

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

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

16. Regular Iterative reconstruction method 16 Interpolation method

17. 13 9 3 4 1 2 17

18. Regular LCB Method Iterative Super-wide LCB Method

19. RTM Method Iterative Super-wide LCB Method

20. RTM Method Iterative 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

23. Thanks！ 23