Dedicated to the Memory of Ding LeeNo Access

Reverse Time Migration Using the Pseudospectral Time-Domain Algorithm

Institute of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen, 361005, P. R. China

E-mail Address: swordshadow@foxmail.com

Search for more papers by this author

Zichao Guo

Institute of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen, 361005, P. R. China

E-mail Address: gzc@xmu.edu.cn

Search for more papers by this author

Hai Liu

Institute of Electromagnetics and Acoustics, Department of Electronic Science, Xiamen University, Xiamen, 361005, P. R. China

E-mail Address: liuhai8619@xmu.edu.cn

Search for more papers by this author

, and

Qing Huo Liu

Department of Electrical and Computer Engineering, Duke University, Durham, NC, 27708, USA

E-mail Address: qhliu@duke.edu

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218396X16500053Cited by:10 (Source: Crossref)

Abstract

We propose a pre-stack reverse time migration (RTM) seismic imaging method using the pseudospectral time-domain (PSTD) algorithm. Traditional pseudospectral method uses the fast Fourier transform (FFT) algorithm to calculate the spatial derivatives, but is limited by the wraparound effect due to the periodicity assumed in the FFT. The PSTD algorithm combines the pseudospectral method with a perfectly matched layer (PML) for acoustic waves. PML is a highly effective absorbing boundary condition that can eliminate the wraparound effect. It enables a wide application of the pseudospectral method to complex models. RTM based on the PSTD algorithm has advantages in the computational efficiency compared to traditional methods such as the second-order and high order finite difference time-domain (FDTD) methods. In this work, we implement the PSTD algorithm for acoustic wave equation based RTM. By applying the PSTD-RTM method to various seismic models and comparing it with RTM based on the eighth-order FDTD method, we find that PSTD-RTM method has better performance and saves more than 50% memory. The method is suitable for parallel computation, and has been accelerated by general purpose graphics processing unit.

Keywords:

References

1. W. W. Symes, Reverse time migration with optimal checkpointing, Geophysics 72(5) (2007) SM213–SM221. Crossref, Web of Science, Google Scholar
2. A. C. Lesage, H. B. Zhou, F. Ortigosa, M. A. Polo and J. M. Cela, 3D reverse-time migration with hybrid finite difference-pseudospectral method, in 2008 SEG Annual Meeting (Society of Exploration Geophysicists, 2008). Google Scholar
3. Y. Wang, Reverse-time migration by a variable time-step and space-grid method, SEG Technical Program Expanded Abstracts 19(1) (1999) 798–801. Google Scholar
4. J. Etgen and M. J. OBrien, Computational methods for large-scale 3D acoustic finite-difference modeling: A tutorial, Geophysics 72(5) (2007) SM223–SM230. Crossref, Web of Science, Google Scholar
5. R. G. Clapp, Reverse time migration: Saving the boundaries, Stanford Exploration Project (2008) 136. Google Scholar
6. R. G. Clapp, Reverse time migration with random boundaries, SEG Expanded Abstracts 28(1) (2009) 2809–2813. Google Scholar
7. H. Guan, Y. C. Kim, J. Ji, K. W. Yoon, B. Wang, W. Xu and Z. Li, Multistep reverse time migration, The Leading Edge 28(4) (2009) 442–447. Crossref, Google Scholar
8. E. Baysal, D. D. Kosloff and J. W. C. Sherwood, Reverse time migration, Geophysics 48(11) (1983) 1514–1524. Crossref, Web of Science, Google Scholar
9. B. Fornberg, The pseudospectral method: Comparisons with finite differences for the elastic wave equation, Geophysics 52(4) (1987) 483–501. Crossref, Web of Science, Google Scholar
10. G. Zhan, R. C. Pestana and P. L. Stoffa, An efficient hybrid pseudospectral/finite-difference scheme for solving the TTI pure P-wave equation, J. Geophys. Eng. 10(2) (2013) 25004–25011. Crossref, Web of Science, Google Scholar
11. X. Du, J. C. Bancroft and L. R. Lines, Anisotropic reverse-time migration for tilted TI media, Geophys. Prospect. 55 (6) (2007) 853–869. Crossref, Web of Science, Google Scholar
12. W. B. Sun and Z. D. Sun, VSP reverse time migration based on the pseudo-spectral method and its applications, Chinese J. Geophys. 53(9) (2010) 2196–2203. Web of Science, Google Scholar
13. R. C. Pestana, B. Ursin and P. L. Stoffa, Rapid expansion and pseudo spectral implementation for reverse time migration in VTI media, J. Geophys. Eng. 9(3) (2012) 291–301. Crossref, Web of Science, Google Scholar
14. Y. Kim, Y. Cho, U. Jang and C. Shin, Acceleration of stable TTI P-wave reverse-time migration with GPUs, Comput. Geosci. 52 (2013) 204–217. Crossref, Web of Science, Google Scholar
15. C. Cerjan, D. Kosloff, R. Kosloff and M. Reshef, A nonreflecting boundary condition for discrete acoustic and elastic wave equations, Geophysics 50(4) (1985) 705–708. Crossref, Web of Science, Google Scholar
16. Q. H. Liu, Large-scale simulations of electromagnetic and acoustic measurements using the pseudospectral time-domain (PSTD) algorithm, IEEE Trans. Geosci. Remote Sensing 37(2) (1999) 917–926. Crossref, Web of Science, Google Scholar
17. M. H. Karazincir and C. M. Gerrard, Explicit high-order reverse time pre-stack depth migration, in the 10th Int. Cong. of the Brazilian Geophysical Society, Society of Exploration Geophysicists (2007). Google Scholar
18. C. Spa, A. Garriga and J. Escolano, Impedance boundary conditions for pseudo-spectral time-domain methods in room acoustics, Appl. Acoust. 71(5) (2010) 402–410. Crossref, Web of Science, Google Scholar
19. Q. H. Liu, The PSTD algorithm: A time-domain method requiring only two cells per wavelength, Microw. Opt. Technol. Lett. 15(3) (1997) 158–165. Crossref, Web of Science, Google Scholar
20. Q. H. Liu, The pseudospectral time-domain (PSTD) algorithm for acoustic waves in absorptive media, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 45(4) (1998) 1044–1055. Crossref, Web of Science, Google Scholar
21. Q. H. Liu and J. P. Tao, The perfectly matched layer for acoustic waves in absorptive media, J. Acoust. Soc. Am. 102(4) (1997) 2072–2082. Crossref, Web of Science, Google Scholar
22. Q. H. Liu, Some current trends in numerical methods for transient acoustic and elastic waves in multidimensional inhomogeneous media, Current Topics Acoust. Res. 2 (1998) 31–42. Google Scholar
23. Y. Q. Zeng and Q. H. Liu, A multidomain PSTD method for 3D elastic wave equations, Bull. Seismol. Soc. Am. 94(3) (2004) 1002–1015. Crossref, Web of Science, Google Scholar
24. Y. H. Ha, K. Lee and W. Seong, Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method, Numer. Algor. 54(3) (2010) 411–430. Crossref, Web of Science, Google Scholar
25. M. Hornikx, R. Waxler and J. Forss, The extended Fourier pseudospectral time-domain method for atmospheric sound propagation, J. Acoust. Soc. Am. 128(4) (2010) 1632. Crossref, Web of Science, Google Scholar
26. M. Hornikx, W. D. Roeck, T. Toulorge and W. Desmet, Flow and geometrical effects on radiated noise from exhaust pipes computed by the Fourier pseudospectral time-domain method, Comput. Fluids 116 (2014) 176–191. Crossref, Web of Science, Google Scholar
27. J. L. Robertson, B. T. Cox and B. E. Treeby, Quantifying numerical errors in the simulation of transcranial ultrasound using pseudospectral methods, in 2014 IEEE Int. Ultrasonics Symp. Proc., Institute of Electrical and Electronic Engineers (2014) 2000–2003. Google Scholar
28. G. Ye, C. H. Deng and Q. H. Liu, The PSTD method with the 4th-order time integration for 3D TAT reconstruction of a breast model, J. Comput. Acoust. 22(4) (2014) 325–325. Link, Web of Science, Google Scholar
29. C. Spa, J. Escolano and A. Garriga, Semiempirical boundary conditions for the linearized acoustic Euler equations using pseudo-spectral time-domain methods, Appl. Acoust. 72(4) (2011) 226–230. Crossref, Web of Science, Google Scholar
30. C. Spa, P. Reche-Lpez and E. Hernndez, Numerical absorbing boundary conditions based on a damped wave equation for pseudospectral time-domain acoustic simulations, Sci. World J. (2014) 285945. Google Scholar
31. Y. F. Li, O. B. Matar, B. S. Li and X. Chen, Pseudo-spectral simulation of 1D nonlinear propagation in heterogeneous elastic media, Wave Motion 52 (2015) 54–65. Crossref, Web of Science, Google Scholar
32. J. P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114(2) (1994) 185–200. Crossref, Web of Science, Google Scholar
33. W. C. Chew and Q. H. Liu, Perfectly matched layers for elastodynamics: A new absorbing boundary condition, J. Comput. Acoust. 4(4) (1996) 341–359. Link, Web of Science, Google Scholar
34. W. Hu, A. Abubakar and T. Habashy, Application of the nearly perfectly matched layer in acoustic wave modeling, Geophysics 72(5) (2007) SM169–SM175. Crossref, Web of Science, Google Scholar
35. T. Hagstrom, A new construction of perfectly matched layers for hyperbolic systems with applications to the linearized euler equations, Math. Numer. Aspects Wave Propag. 1 (2003) 125–129. Google Scholar
36. R. Martin, D. Komatitsch, S. D. Gedney and E. Bruthiaux, A high-order time and space formulation of the unsplit perfectly matched layer for the seismic wave equation using auxiliary differential equations (ADE-PML), Comput. Model. Eng. Sci. 56(1) (2010) 17–41. Web of Science, Google Scholar
37. Y. Q. Zeng, Q. H. Liu and G. Zhao, Multidomain pseudospectral time-domain (PSTD) method for acoustic waves in lossy media, J. Comput. Acoust. 12(3) (2004) 277–299. Link, Web of Science, Google Scholar
38. F. H. Drossaert and A. Giannopoulos, A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves, Geophysics 72(2) (2007) T9–T17. Crossref, Web of Science, Google Scholar
39. T. Wang and X. M. Tang, Finite-difference modeling of elastic wave propagation: A nonsplitting perfectly matched layer approach, Geophysics 68(5) (2003) 1749–1755. Crossref, Web of Science, Google Scholar
40. D. Komatitsch and R. Martin, An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation, Geophysics 72(5) (2007) SM155–SM167. Crossref, Web of Science, Google Scholar
41. S. D. Gedney and Bo Zhao, An auxiliary differential equation formulation for the Complex-Frequency Shifted PML, IEEE Trans. Antennas & Propag. 58(3) (2010) 838–847. Crossref, Web of Science, Google Scholar
42. W. Zhang and Y. Shen, Unsplit complex frequency shifted PML implementation using auxiliary differential equation for seismic wave modeling, Geophysics 75(4) (2010) T141–T154. Crossref, Web of Science, Google Scholar
43. R. Q. Shi, S. X. Wang and J. G. Zhao, An unsplit complex-frequency-shifted PML based on matched Z-transform for FDTD modelling of seismic wave equations, J. Geophys. Eng. 9(2) (2012) 218–229. Crossref, Web of Science, Google Scholar
44. N. X. Feng, Y. Q. Yue and Q. H. Liu, Direct $Z$ $Z$ -transform implementation of the cfs-pml based on memory-minimized method, IEEE Trans. Microw. Theory Tech. 63(3) (2015) 877–882. Crossref, Web of Science, Google Scholar
45. NVIDIA Corporation, NVIDIA CUDA ZONE, https://developer.nvidia.com/cuda-zone (accessed on 21 January 2015). Google Scholar
46. R. G. Clapp, H. H. Fu and O. Lindtjorn, Selecting the right hardware for reverse time migration, The Leading Edge 29(1) (2010) 48–58. Crossref, Google Scholar
47. X. H. Shi, C. Li, S. H. Wang and X. Wang, Computing prestack kirchhoff time migration on general purpose $g p u$ $g p u$ , Comput. Geosci. 37(10) (2011) 1702–1710. Crossref, Web of Science, Google Scholar
48. G. F. Liu, X. Hong Meng and L. Hong, Accelerating finite difference wavefield-continuation depth migration by GPU, Appl. Geophys. 9(1) (2012) 41–48. Crossref, Web of Science, Google Scholar
49. P. Micikevicius, 3D finite difference computation on GPUs using CUDA, in Proc. 2nd Workshop on General Purpose Processing on Graphics Processing Units (ACM, 2009), pp. 79–84. Google Scholar
50. W. Liu, T. Nemeth, A. Loddoch, J. Stefani, R. Ergas, L. Zhuo, B. Volz, O. Pell and J. Huggett, Anisotropic reverse-time migration using co-processors, in the 79th Society of Exploration Geophysicists (SEG) (Houston, 2009). Google Scholar
51. O. Holberg, Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena, Geophys. Prospect. 35(6) (1987) 629–655. Crossref, Web of Science, Google Scholar
52. W. J. Wu, L. R. Lines and H. X. Lu, Analysis of higher-order, finite-difference schemes in 3-D reverse-time migration, Geophysics 61(3) (1996) 845–856. Crossref, Web of Science, Google Scholar
53. M. Kindelan, A. Kamel and P. Sguazzero, On the construction and efficiency of staggered numerical differentiators for the wave equation, Geophysics 55(1) (1990) 107–110. Crossref, Web of Science, Google Scholar
54. E. Dussaud, W. W. Symes, P. Williamson, L. Lemaistre, P. Singer, B. Denel and A. Cherrett, Computational strategies for reverse-time migration, in 2008 SEG Annual Meeting (Society of Exploration Geophysicists, 2008). Google Scholar
55. S. Chattopadhyay and G. A McMechan, Imaging conditions for prestack reverse-time migration, Geophysics 73(3) (2008) S81–S89. Crossref, Web of Science, Google Scholar
56. G. S. Martin, R. Wiley and K. J. Marfurt, Marmousi2: An elastic upgrade for marmousi, The Leading Edge 25(2) (2006) 156–166. Crossref, Google Scholar
57. G. H. F. Gardner, L. W. Gardner and A. R. Gregory, Formation velocity and density-the diagnostic basics for stratigraphic traps, Geophysics 39(6) (1974) 770–780. Crossref, Web of Science, Google Scholar
58. Ö. Yilmaz and S. M. Doherty, Seismic Data Processing, Vol. 2 (Society of Exploration Geophysicists, Tulsa, 1987). Google Scholar
59. J. C. Lecomte, E. Campbell and J. Letouzey, Building the SEG/EAEG overthrust velocity macro model, in EAEG/SEG Summer Workshop-Construction of 3-D Macro Velocity-Depth Models, European Association of Exploration Geophysicists (1994). Google Scholar