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A simplified calculation for adaptive coefficients of finite-difference frequency-domain method

  • Seismic modeling and wave propagation
  • Published:
Applied Geophysics Aims and scope Submit manuscript

Abstract

The finite-difference frequency domain (FDFD) method is widely applied for simulating seismic wavefields, and a key to achieving successful FDFD simulation is to construct FDFD coefficients that can effectively suppress numerical dispersion. Among the existing FDFD coefficients for seismic wavefield simulation, adaptive FDFD coefficients that vary with the number of wavelengths per grid can suppress numerical dispersion to the maximum extent. The current methods for calculating adaptive FDFD coefficients involve numerical integration, conjugate gradient (CG) optimization, sequential initial value selection, and smooth regularization, which are difficult to implement and inefficient in calculations. To simplify the calculation of adaptive FDFD coefficients and improve the corresponding computational efficiency, this paper proposes a new method for calculating adaptive FDFD coefficients. First, plane-wave solutions with different discrete propagation angles are substituted in the FDFD scheme, and the corresponding least-squares problem is constructed. As this problem is ill-conditioned and obtaining smooth adaptive FDFD coefficients by the conventional solving method based on normal equations is difficult, this paper proposes solving the least-squares problem by solving the corresponding overdetermined linear system of equations through QR matrix decomposition. Compared with the existing methods for calculating adaptive FDFD coefficients based on numerical integration, CG optimization, and sequential initial value selection, the proposed method allows for a simplified computational process and considerably higher computational efficiency. Numerical wavefield simulation results show that the adaptive-coefficient FDFD method based on QR matrix decomposition can achieve the same accuracy as those based on numerical integration, CG optimization, and sequential initial value selection while requiring less computation time.

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References

  • Aghamiry, H. S., Gholami, A., Combe, L., and Operto, S. 2022, Accurate 3D frequency-domain seismic wave modeling with the wavelength-adaptive 27-point finite-difference stencil: A tool for full-waveform inversion: Geophysics, 87(3), R305–R324.

    Article  Google Scholar 

  • Chen, J. B., 2012, An average-derivative optimal scheme for frequency-domain scalar wave equation: Geophysics, 77(6), T201–T210.

    Article  Google Scholar 

  • Chen, J. B., 2014, A 27-point scheme for a 3D frequency-domain scalar wave equation based on an average-derivative method: Geophysical Prospecting, 62(2), 258–277.

    Article  Google Scholar 

  • Dablain, M. A., 1986, The application of high-order differencing to the scalar wave equation: Geophysics, 51(1), 54–66.

    Article  Google Scholar 

  • Fan, N., Zhao, L. F., Xie, X. B., Tang, X. G., and Yao, Z. X., 2017, A general optimal method for a 2D frequency-domain finite-difference solution of scalar wave equation: Geophysics, 82(3), T121–T132.

    Article  Google Scholar 

  • Gao, J., Xu, W., Wu, B., Li, B., and Zhao, H., 2018, Trapezoid grid finite difference seismic wavefield simulation with uniform depth sampling interval: Chinese Journal of Geophysics, 61(8), 3285–3296.

    Google Scholar 

  • Golub, G. H., and Van Loan, C. F., 2013, Matrix computations: Johns Hopkins University Press, Baltimore, US, 233–302.

    Book  Google Scholar 

  • Harris, C. R., Millman, K. J., Van Der Walt, S. J., et al., 2020, Array programming with NumPy: Nature, 585(7825), 357–362.

    Article  CAS  Google Scholar 

  • Jo, C. H., Shin, C., and Suh, J. H., 1996, An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator: Geophysics, 61(2), 529–537.

    Article  Google Scholar 

  • Komatitsch, D., and Martin, R., 2007, An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation: Geophysics, 72(5), SM155–SM167.

    Article  Google Scholar 

  • Liu, Q. H., and Tao, J., 1997, The perfectly matched layer for acoustic waves in absorptive media: The Journal of the Acoustical Society of America, 102(4), 2072–2082.

    Article  Google Scholar 

  • Min, D. J., Shin, C., Kwon, B. D., and Chung, S., 2000, Improved frequency-domain elastic wave modeling using weighted-averaging difference operators: Geophysics, 65(3), 884–895.

    Article  Google Scholar 

  • Operto, S., Amestoy, P., Aghamiry, H., et al., 2023, Is 3D frequency-domain FWI of full-azimuth/long-offset OBN data feasible? The Gorgon data FWI case study: The Leading Edge, 42(3), 173–183.

    Article  Google Scholar 

  • Petra, C. G., Schenk, O., and Anitescu, M., 2014, Realtime stochastic optimization of complex energy systems on high-performance computers: Computing in Science and Engineering, 16(5), 32–42.

    Article  Google Scholar 

  • Petra, C. G., Schenk, O., Lubin, M., and Gärtner, K., 2014, An augmented incomplete factorization approach for computing the Schur complement in stochastic optimization: SIAM Journal on Scientific Computing, 36(2), C139–C162.

    Article  Google Scholar 

  • Pratt, R. G., 1990, Frequency-domain elastic wave modeling by finite differences: A tool for crosshole seismic imaging: Geophysics, 55(5), 626–632.

    Google Scholar 

  • Pratt, R. G., and Worthington, M. H., 1990, Inverse theory applied to multi-source cross-hole tomography, Part I: acoustic wave-equation method: Geophysical Prospecting, 38(3), 287–310.

    Google Scholar 

  • Ren, Z., and Liu, Y., 2013, A hybrid absorbing boundary condition for frequency-domain finite-difference modelling: Journal of Geophysics and Engineering, 10(5), 054003, doi: https://doi.org/10.1088/1742-2132/10/5/054003.

    Article  Google Scholar 

  • Rupp, K., Tillet, P., Rudolf, F., et al., 2016, ViennaCL—linear algebra library for multi-and many-core architectures: SIAM Journal on Scientific Computing, 38(5), S412–S439.

    Article  Google Scholar 

  • Shin, C., and Sohn, H., 1998, A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operator: Geophysics, 63(1), 289–296.

    Article  Google Scholar 

  • Štekl, I., and Pratt, R. G., 1998, Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators: Geophysics, 63(5), 1779–1794.

    Article  Google Scholar 

  • Tournier, P. H., Jolivet, P., Dolean, V., Aghamiry, H. S., Operto, S., and Riffo, S., 2022, 3D finite-difference and finite-element frequency-domain wave simulation with multilevel optimized additive Schwarz domain-decomposition preconditioner: A tool for full-waveform inversion of sparse node data sets: Geophysics, 87(5), T381–T402.

    Google Scholar 

  • Virieux, J., 1986, P-SV wave propagation in heterogeneous media: Velocity-stress finite-difference method: Geophysics, 51(4), 889–901.

    Article  Google Scholar 

  • Wang, E., Ba, J., and Liu, Y., 2018, Time-space-domain implicit finite-difference methods for modeling acoustic wave equations: Geophysics, 83(4), T175–T193.

    Article  Google Scholar 

  • Xu, J., Hu, H., Zhan, Q., Zhong, Y., and Liu, Q. H., 2023, Nearly perfectly matched layer implementation for time domain spectral element modelling of wave propagation in 3D heterogeneous and anisotropic porous media: Journal of Applied Geophysics, 208, 104870, doi: https://doi.org/10.1016/j.jappgeo.2022.104870.

    Article  Google Scholar 

  • Xu, W., and Gao, J., 2018, Adaptive 9-point frequency-domain finite difference scheme for wavefield modeling of 2D acoustic wave equation: Journal of Geophysics and Engineering, 15(4), 1432–1445.

    Article  Google Scholar 

  • Xu, W., Zhong, Y., Wu, B., Gao, J., and Liu, Q. H., 2021a, Adaptive complex frequency with V-cycle GMRES for preconditioning 3D Helmholtz equation: Geophysics, 86(5), T349–T359.

    Article  Google Scholar 

  • Xu, W., Wu, B., Zhong, Y., Gao, J., and Liu, Q. H., 2021b, Adaptive 27-point finite-difference frequency-domain method for wave simulation of 3D acoustic wave equation: Geophysics, 86(6), T439–T449.

    Article  Google Scholar 

  • Zhang, H., Liu, H., Lui, L., Jin, W., and Shi, X., 2014, Frequency domain acoustic equation high-order modeling based on an average-derivative method: Chinese Journal of Geophysics, 57(5), 1599–1611.

    Google Scholar 

  • Zhang, J. H., and Yao, Z. X., 2013, Optimized finite-difference operator for broadband seismic wave modeling: Geophysics, 78(1), A13–A18.

    Article  Google Scholar 

  • Zhang, Y., Gao, J., and Peng, J., 2018, Variable-order finite difference scheme for numerical simulation in 3-D poroelastic media: IEEE Transactions on Geoscience and Remote Sensing, 56(5), 2991–3001.

    Article  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 42174161, No. 41974123), China Postdoctoral Science Foundation (No. 2022M711004), China National Petroleum Corporation Exploration and Development Research Institute Open Fund (No. 822102016), and the Jiangsu Province Science Fund for Distinguished Young Scholars (No. BK20200021).

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Authors and Affiliations

  1. School of Earth Science and Engineering, Hohai University, Nanjing, Jiangsu, 211000, China

    Wen-Hao Xu, Jing Ba & José Maria Carcione

  2. National Institute of Oceanography and Applied Geophysics (OGS), Sgonico, Trieste, 34010, Italy

    José Maria Carcione

  3. China National Petroleum Corporation Exploration and Development Research Institute, Beijing, 100083, China

    Zhi-Fang Yang & Xin-Fei Yan

  4. China National Petroleum Corporation Key Laboratory of Geophysics, Beijing, 100083, China

    Zhi-Fang Yang & Xin-Fei Yan

Authors
  1. Wen-Hao Xu
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  2. Jing Ba
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  3. José Maria Carcione
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  4. Zhi-Fang Yang
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  5. Xin-Fei Yan
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Corresponding author

Correspondence to Jing Ba.

Additional information

Xu Wen-Hao received his Ph.D. degree in Information and Communication Engineering from Xi’an Jiaotong University in 2022 and has been working as an assistant researcher in the School of Earth Science and Engineering, Hohai University, since 2022. His main research interests are seismic wavefield simulation and seismic parameter inversion.

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Xu, WH., Ba, J., Carcione, J.M. et al. A simplified calculation for adaptive coefficients of finite-difference frequency-domain method. Appl. Geophys. 20, 262–277 (2023). https://doi.org/10.1007/s11770-023-1045-8

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  • DOI: https://doi.org/10.1007/s11770-023-1045-8

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