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ISSN  2096-3955

CN  10-1502/P

Citation: Zhang, D. Y., Pan, W. Y., Yang, D. H., Qiu, L. Y., Dong, X. P. and Meng, W. J. (2021). Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method. Earth Planet. Phys., 5(2), 149–157doi: 10.26464/epp2021022

2021, 5(2): 149-157. doi: 10.26464/epp2021022


Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method


Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China


Key Laboratory of Petroleum Resource Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China


Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

Corresponding author: DingHui Yang,

Received Date: 2020-09-30
Web Publishing Date: 2021-03-01

The nearly analytic discrete (NAD) method is a kind of finite difference method with advantages of high accuracy and stability. Previous studies have investigated the NAD method for simulating wave propagation in the time-domain. This study applies the NAD method to solving three-dimensional (3D) acoustic wave equations in the frequency-domain. This forward modeling approach is then used as the “engine” for implementing 3D frequency-domain full waveform inversion (FWI). In the numerical modeling experiments, synthetic examples are first given to show the superiority of the NAD method in forward modeling compared with traditional finite difference methods. Synthetic 3D frequency-domain FWI experiments are then carried out to examine the effectiveness of the proposed methods. The inversion results show that the NAD method is more suitable than traditional methods, in terms of computational cost and stability, for 3D frequency-domain FWI, and represents an effective approach for inversion of subsurface model structures.

Key words: three-dimension; frequency-domain; NAD method; forward modeling; full waveform inversion

Bozdağ, E., Peter, D., Lefebvre, M., Komatitsch, D., Tromp, J., Hill, J., Podhorszki, N., and Pugmire, D. (2016). Global adjoint tomography: first-generation model. Geophys. J. Int., 207(3), 1739–1766.

Brossier, R., Operto, S., and Virieux, J. (2009). Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion. Geophysics, 74(6), WCC105–WCC118.

Day, S. M. (1982). Three-dimensional simulation of spontaneous rupture: the effect of nonuniform prestress. Bull. Seismol. Soc. Am., 72(6A), 1881–1902.

Dong, X. P., Yang, D. H., and Niu, F. L. (2019). Passive Adjoint tomography of the crustal and upper mantle beneath eastern Tibet with a W2-norm misfit function. Geophys. Res. Lett., 46(22), 12986–12995.

He, X. J., Yang, D. H., Huang, X. Y., and Ma, X. (2020). A numerical dispersion-dissipation analysis of discontinuous Galerkin methods based on quadrilateral and triangular elements. Geophysics, 85(3), T101–T121.

Igel, H., Mora, P., and Riollet, B. (1995). Anisotropic wave propagation through finite-difference grids. Geophysics, 60(4), 1203–1216.

Kelly, K. R., Ward, R. W., Treitel, S., and Alford, R. M. (1976). Synthetic seismograms: a finite-difference approach. Geophysics, 41(1), 2–27.

Komatitsch, D., and Tromp, J. (2003). A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation. Geophys. J. Int., 154(1), 146–153.

Komatitsch, D., Tsuboi, S., and Tromp, J. (2005). The spectral-element method in seismology. In A. Levander, et al. (Eds.), Seismic Earth: Array Analysis of Broadband Seismograms, Volume 157 (pp. 205-227). Washington, DC: American Geophysical Union.

Kosloff, D. D., and Baysal, E. (1982). Forward modeling by a Fourier method. Geophysics, 47(10), 1402–1412.

Lailly, P. (1983). The seismic inverse problem as a sequence of before stack migration. In SIAM Conference on Inverse Scattering: Theory and Applications, Expanded Abstracts (pp. 206-220). SIAM.222

Lang, C., and Yang, D. H. (2017). A nearly analytic discrete method for solving the acoustic-wave equations in the frequency domain. Geophysics, 82(1), T43–T57.

Li, J. S., Yang, D. H., and Liu, F. Q. (2013). An efficient reverse time migration method using local nearly analytic discrete operator. Geophysics, 78(1), S15–S23.

Liu, Q. Y., and Tromp, J. (2006). Finite-frequency kernels based on adjoint methods. Bull. Seismol. Soc. Am., 96(6), 2383–2397.

Liu, S. L., Li, X. F., Wang, W. S., Liu, Y. S., Zhang, M. G., and Zhang, H. (2014). A new kind of optimal second-order symplectic scheme for seismic wave simulations. Sci China Earth Sci, 57(4), 751–758.

Liu, S. L., Li, X. F., Wang, W. S., Xu, L., and Li, B. F. (2015). A modified symplectic scheme for seismic wave modeling. J. Appl. Geophys., 116, 110–120.

Lysmer, J., and Drake, L. A. (1972). A finite element method for seismology. Methods Comput. Phys.: Adv. Res. Appl., 11, 181–216.

Ma, X., Yang, D. H., He, X. J., Li, J. S., and Zheng, Y. C. (2018). A new high-order scheme based on numerical dispersion analysis of the wave phase velocity for semidiscrete wave equations. Geophysics, 83(3), T123–T138.

Marfurt, K. J. (1984). Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations. Geophysics, 49(5), 533–549.

Mora, P. (1987). Nonlinear two-dimensional elastic inversion of multioffset seismic data. Geophysics, 52(9), 1211–1228.

Nocedal, J., and Wright, S. J. (2006). Numerical Optimization (2nd ed). Berlin: Springer.222

Operto, S., Ravaut, C., Improta, L., Virieux, J., Herrero, A., and Dell’Aversana, P. (2004). Quantitative imaging of complex structures from dense wide-aperture seismic data by multiscale traveltime and waveform inversions: a case study. Geophys. Prosp., 52(6), 625–651.

Operto, S., Virieux, J., Amestoy, P., L’Excellent, J. Y., Giraud., L., and Ben Hadj Ali, H. (2007). 3-D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: a feasibility study. Geophysics, 72(5), SM195–SM211.

Pan, W. Y., Innanen, K. A., Margrave, G. F., Fehler, M. C., Fang, X. D., and Li, J. X. (2016). Estimation of elastic constants for HTI media using Gauss-Newton and full-Newton multiparameter full-waveform inversion. Geophysics, 81(5), R275–R291.

Pan, W. Y., Innanen, K. A., and Liao, W. Y. (2017). Accelerating Hessian-free Gauss-Newton full-waveform inversion via l-BFGS preconditioned conjugate-gradient algorithm. Geophysics, 82(2), R49–R64.

Plessix, R. E. (2009). Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics, 74(6), WCC149–WCC157.

Pratt, R. G., and Worthington, M. H. (1990). Inverse theory applied to multi-source cross-hole tomography. Part 1: acoustic wave-equation method. Geophys. Prosp., 38(3), 287–310.

Pratt, R. G., Shin, C., and Hick, G. J. (1998). Gauss–Newton and full Newton methods in frequency–space seismic waveform inversion. Geophys. J. Int., 133(2), 341–362.

Pratt, R. G. (1999). Seismic waveform inversion in the frequency domain, Part 1: theory and verification in a physical scale model. Geophysics, 64(3), 888–901.

Seriani, G., and Priolo, E. (1994). Spectral element method for acoustic wave simulation in heterogeneous media. Finite Elem. Anal. Des., 16(3-4), 337–348.

Sirgue, L., and Pratt, R. G. (2004). Efficient waveform inversion and imaging: a strategy for selecting temporal frequencies. Geophysics, 69(1), 231–248.

Song, Z. M., and Williamson, P. R. (1995). Frequency-domain acoustic-wave modeling and inversion of crosshole data: part I-2.5-D modeling method. Geophysics, 60(3), 784–795.

Tape, C., Liu, Q. Y., and Tromp, J. (2007). Finite-frequency tomography using adjoint methods-methodology and examples using membrane surface waves. Geophys. J. Int., 168(3), 1105–1129.

Tarantola, A. (1986). A strategy for nonlinear elastic inversion of seismic reflection data. Geophysics, 51(10), 1893–1903.

Tong, P., Yang, D. H., and Hua, B. L. (2011). High accuracy wave simulation–revised derivation, numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation. International Journal of Solids and Structures, 48(1), 56–70.

Tong, P., Yang, D. H., Hua, B. L., and Wang, M. X. (2013). A high-order stereo-modeling method for solving wave equations. Bull. Seismol. Soc. Am., 103(2A), 811–833.

Tromp, J., Tape, C., and Liu, Q. (2005). Seismic tomography, adjoint methods, time reversal, and banana-doughnut kernels. Geophys. J. Int., 160(1), 195–216.

Virieux, J., and Operto, S. (2009). An overview of full-waveform inversion in exploration geophysics. Geophysics., 74(6), WCC1–WCC26.

Wang, J., Yang, D. H., Jing H., and Wu, H. (2019). Full waveform inversion based on the ensemble Kalman filter method using uniform sampling without replacement. Sci. Bull., 64(5), 321–330.

Yang, D. H., Teng, J. W., Zhang, Z. J., and Liu, E. R. (2003). A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media. Bull. Seismol. Soc. Am., 93(2), 882–890.

Yang, D. H., Lu, M., Wu, R. S., and Peng, J. M. (2004). An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations. Bull. Seismol. Soc. Am., 94(5), 1982–1992.

Yang, D. H., Peng, J. M., Lu, M., and Terlaky, T. (2006). Optimal nearly analytic discrete approximation to the scalar wave equation. Bull. Seismol. Soc. Am., 96(3), 1114–1130.

Yang, D. H., Wang, L., and Deng, X. Y. (2010). An explicit split-step algorithm of the implicit Adams method for solving 2-D acoustic and elastic wave equations. Geophys. J. Int., 180(1), 291–310.

Yang, D. H., Wang, N., and Liu, E. (2012). A strong stability-preserving predictor-corrector method for the simulation of elastic wave propagation in anisotropic media. Commun. Comput. Phys., 12(4), 1006–1032.


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Three-dimensional frequency-domain full waveform inversion based on the nearly-analytic discrete method

DeYao Zhang, WenYong Pan, DingHui Yang, LingYun Qiu, XingPeng Dong, WeiJuan Meng