Abstract
Frequency-domain numerical simulation is the most important foundation of frequency-domain full-waveform inversion and reverse time migration. The accuracy of numerical simulation seriously affects the results of the seismic inversion and image. In this article, we develop an optimized compact finite difference scheme for acoustic wave equation in frequency domain to improve numerical simulation accuracy. For the sake of avoiding the extra memory and computational costs caused by solving the inverse of a pentadiagonal band matrix, we calculate the optimized compact finite difference discrete operator for the Laplace operator in the numerical simulation. Although the optimized compact finite difference scheme has only second-order formal accuracy, it has a spectral-like resolution feature. This method can significantly reduce the numerical dispersion and the numerical anisotropy. We find that the results of the optimized compact finite difference scheme agree well with the analytic solution according to accuracy analysis. Two numerical simulations are done to verify the theoretical analysis of the optimized compact finite difference scheme.
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References
Alford RM, Kelly KR, Boore DM (1974) Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics 39:834–842. https://doi.org/10.1190/1.1440470
Alterman Z, Karal FC (1968) Propagation of elastic waves in layered media by finite difference methods. Bull Seismol Soc Am 58:367–398
Baysal E, Dan DK, Sherwood JWC (1983) Reverse time migration. Geophysics 48:1514–1524
Berenger JP (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185–200
Chang WF, McMechan GA (1987) Elastic reverse-time migration. Geophysics 52:1365–1375. https://doi.org/10.1190/1.1442249
Chen JB (2012) An average-derivative optimal scheme for frequency-domain scalar wave equation. Geophysics 77:T201–T210. https://doi.org/10.1190/geo2011-0389.1
Chen JB (2014) A 27-point scheme for a 3D frequency-domain scalar wave equation based on an average-derivative method. Geophys Prospect 62:258–277
Chen JB, Cao J (2016) Modeling of frequency-domain elastic-wave equation with an average-derivative optimal method. Geophysics 81:T339–T356. https://doi.org/10.1190/geo2016-0041.1
Chu C, Stoffa PL (2012) An implicit finite-difference operator for the Helmholtz equation. Geophysics 77:97
Hustedt B, Operto S, Virieux J (2004) Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling. Geophys J R Astron Soc 157:1269–1296
Jo C, Shin C, Suh JH (1996) An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator. Geophysics 61:529–537
Kim JW, Lee DJ (1996) Optimized compact finite difference schemes with maximum resolution. J AIAA 34:887–893
Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103:16–42
Liu Y, Sen MK (2009) A practical implicit finite-difference method: examples from seismic modelling. J Geophys Eng 6:231
Lysmer J, Drake LA (1972) A finite element method for seismology. Methods Comput Phys Adv Res Appl 11:181–216
Marfurt KJ (1984) Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics 49:533–549
Marfurt KJ, Shin CS (1989) The future of iterative modeling in geophysical exploration. Handb Geophys Explor Seism Explor 21:203–228
McMechan GA (1983) Migration by extrapolation of time-dependent boundary values. Geophys Prospect 31:413–420
Mora P (1987) Nonlinear two-dimensional elastic inversion of multioffset seismic data. Geophysics 52:1211–1228
Operto S, Virieux J, Amestoy P, Giraud L, L’Excellent JY (2007) 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: a feasibility study. Geophysics 72:195
Pratt RG (1990) Frequency-domain elastic wave modeling by finite differences: a tool for crosshole seismic imaging. Geophysics 55:626–632
Pratt RG, Worthington MH (1990) Inverse theory applied to multi-source cross-hole tomography. Part I: Acoustic wave-equation method. Geophys Prospect 38:287–310
Pratt RG, Shin C, Hick GJ (1998) Gauss–Newton and full Newton methods in frequency-space seismic waveform inversion. Geophys J Int 133:341–362. https://doi.org/10.1046/j.1365-246X.1998.00498.x
Shi P, Angus D, Nowacki A, Yuan S, Wang Y (2018) Microseismic full waveform modeling in anisotropicmedia with moment tensor implementation. Surv Geophys 39:567–611. https://doi.org/10.1007/s10712-018-9466-2
Shin C, Cha YH (2008) Waveform inversion in the Laplace domain. Geophys J Int 173:922–931
Shin C, Sohn H (1998) A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operator. Geophysics 63:289–296. https://doi.org/10.1190/1.1444323
Tang X, Liu H, Zhang H, Liu L, Wang Z (2015) An adaptable 17-point scheme for high-accuracy frequency-domain acoustic wave modeling in 2D constant density media. Geophysics 80:T211–T221. https://doi.org/10.1190/geo2014-0124.1
Tarantola A (1984) Inversion of seismic reflection data in the acoustic approximation. Geophysics 49:1259–1266
Virieux J (1984) SH-wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 49:1933–1942. https://doi.org/10.1190/1.1441605
Virieux J (1986) P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51:889–901. https://doi.org/10.1190/1.1442147
Virieux J, Operto S (2009) An overview of full-waveform inversion in exploration geophysics. Geophysics 74:WCC1–WCC26
Whitmore ND (1983) Iterative depth migration by backward time propagation. Seg Tech Program Expand Abstr 2:646
Yuan S, Wang S, Sun W, Miao L, Li Z (2014) Perfectly matched layer on curvilinear grid for the second-order seismic acoustic wave equation. Explor Geophys 45:94–104. https://doi.org/10.1071/EG13066
Yuan S, Wang S, Luo Y, Wei W, Wang G (2019) Impedance inversion by using the low-frequency full-waveform inversion result as an a priori model. Geophysics 84:R149–R164. https://doi.org/10.1190/geo2017-0643.1
Zhang Y, Xu S, Tang B, Bai B, Huang Y, Huang T (2010) Angle gathers from reverse time migration. Lead Edge 29:1364–1371. https://doi.org/10.1190/1.3517308
Acknowledgements
This research is supported by National Key Scientific Instrument and Equipment Development Project (Grant No. 2018YFF01013503), National Natural Science Foundation of China (Grant No. 41630319) and Major State Research Development Program of China (Grant No. 2016YFC0601101).
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Li, A., Liu, H. Optimized compact finite difference scheme for frequency-domain acoustic wave equation. Acta Geophys. 67, 1391–1402 (2019). https://doi.org/10.1007/s11600-019-00332-z
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DOI: https://doi.org/10.1007/s11600-019-00332-z