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ISSN  2096-3955

CN  10-1502/P

Citation: Liu, Y. S., Xu, T., Wang, Y. H., Teng, J. W., Badal, J., and Lan, H. Q. (2019). An efficient source wavefield reconstruction scheme using single boundary layer values for the spectral element method. Earth Planet. Phys., 3(4), 342–357.doi: 10.26464/epp2019035

2019, 3(4): 342-357. doi: 10.26464/epp2019035


An efficient source wavefield reconstruction scheme using single boundary layer values for the spectral element method


State Key Laboratory of Lithospheric Evolution, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China


Chinese Academy of Sciences Center for Excellence in Tibetan Plateau Earth Sciences, Beijing 100101, China


Department of Earth Science and Engineering, Imperial College London, SW7 2AZ, UK


Physics of the Earth, Sciences-B, University of Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain

Corresponding author: Tao Xu,

Received Date: 2019-01-28
Web Publishing Date: 2019-07-01

In the adjoint-state method, the forward-propagated source wavefield and the backward-propagated receiver wavefield must be available simultaneously either for seismic imaging in migration or for gradient calculation in inversion. A feasible way to avoid the excessive storage demand is to reconstruct the source wavefield backward in time by storing the entire history of the wavefield in perfectly matched layers. In this paper, we make full use of the elementwise global property of the Laplace operator of the spectral element method (SEM) and propose an efficient source wavefield reconstruction method at the cost of storing the wavefield history only at single boundary layer nodes. Numerical experiments indicate that the accuracy of the proposed method is identical to that of the conventional method and is independent of the order of the Lagrange polynomials, the element type, and the temporal discretization method. In contrast, the memory-saving ratios of the conventional method versus our method is at least N when using either quadrilateral or hexahedron elements, respectively, where N is the order of the Lagrange polynomials used in the SEM. A higher memory-saving ratio is achieved with triangular elements versus quadrilaterals. The new method is applied to reverse time migration by considering the Marmousi model as a benchmark. Numerical results demonstrate that the method is able to provide the same result as the conventional method but with about 1/25 times lower storage demand. With the proposed wavefield reconstruction method, the storage demand is dramatically reduced; therefore, in-core memory storage is feasible even for large-scale three-dimensional adjoint inversion problems.

Key words: spectral element method, source wavefield reconstruction, single boundary layer, memory-saving ratio, adjoint method, reverse time migration

Anderson, J. E., Tan, L. J., and Wang, D. (2012). Time-reversal checkpointing methods for RTM and FWI. Geophysics, 77(4), S93–S103.

Berenger, J. P. (1994). A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys., 114(2), 185–200.

Bozdağ, E., Trampert, J., and Tromp, J. (2011). Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements. Geophys. J. Int., 185(2), 845–870.

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., Gholami, Y., Virieux, J., and Operto, S. (2010). 2D frequency-domain seismic wave modeling in VTI media based on a Hp-adaptive discontinuous Galerkin method. In Proceedings of the 72nd EAGE Conference & Exhibition incorporating SPE EUROPEC. Barcelona, Spain: SEG.222

Capdeville, Y., and Marigo, J. J. (2007). Second order homogenization of the elastic wave equation for non-periodic layered media. Geophys. J. Int., 170(2), 823–838.

Carcione, J. M., and Wang, P. J. (1993). A chebyshev collocation method for the wave equation in generalized coordinates. Comput. Fluid Dyn. J., 2(3), 269–290.

Chaljub, E., Capdeville, Y., and Vilotte, J. P. (2003). Solving elastodynamics in a fluid-solid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids. J. Comput. Phys., 187(2), 457–491.

Chaljub, E., and Valette, B. (2004). Spectral element modelling of three-dimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core. Geophys. J. Int., 158(1), 131–141.

Chen, M., Niu, F. L., Liu, Q. Y., Tromp, J., and Zheng, X. F. (2015). Multiparameter adjoint tomography of the crust and upper mantle beneath East Asia: 1. Model construction and comparisons. J. Geophys. Res., 120(3), 1762–1786.

Clapp, R. G. (2009). Reverse time migration with random boundaries. In Proceedings of the 79th Annual International Meeting, SEG, Expanded Abstracts (pp. 2809-2813). Houston, Texas: SEG.222

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

Dai, W., Wang, X., and Schuster, G. T. (2011). Least-squares migration of multisource data with a deblurring filter. Geophysics, 76(5), R135–R146.

De Basabe, J. D., and Sen, M. K. (2007). Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equation. Geophysics, 72(6), T81–T95.

De Basabe, J. D., and Sen, M. K. (2010). Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping. Geophys. J. Int., 181(1), 577–590.

Dussaud, E., Symes, W. W., Williamson, P., Lemaistre, L., Singer, P., Denel, B., and Cherrett, A. (2008). Computational strategies for reverse-time migration. In Proceedings of the 78th Annual International Meeting, SEG, Expanded Abstracts (pp. 2267-2271). Las Vegas: Society of Exploration Geophysicists.222

Feng, B., and Wang, H. Z. (2012). Reverse time migration with source wavefield reconstruction strategy. J. Geophys. Eng., 9(1), 69–74.

Feng, B., Zhou, Y., and Wang, H. Z. (2013). Reply to comment on ‘Reverse time migration with source wavefield reconstruction strategy’. J. Eng. Geophys., 10(2), 028002.

Fichtner, A., Bunge, H. P., and Igel, H. (2006a). The adjoint method in seismology: I. Theory. Earth Planet. Inter., 157(1-2), 86–104.

Fichtner, A., Bunge, H. P., and Igel, H. (2006b). The adjoint method in seismology—II. Applications: traveltimes and sensitivity functionals. Phys. Earth Planet. Inter., 157(1-2), 105–123.

Fichtner, A., Kennett, B. L. N., Igel, H., and Bunge, H. P. (2009). Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods. Geophys. J. Int., 179(3), 1703–1725.

Fletcher, R. P., and Robertsson, J. O. A. (2011). Time-varying boundary conditions in simulation of seismic wave propagation. Geophysics, 76(1), A1–A6.

Gauthier, O., Virieux, J., and Tarantola, A. (1986). Two-dimensional nonlinear inversion of seismic waveforms; numerical results. Geophysics, 51(7), 1387–1403.

Jund, S., and Salmon, S. (2007). Arbitrary high-order finite element schemes and high-order mass lumping. Int. J. Appl. Math. Comput. Sci., 17(3), 375–393.

Komatitsch, D., and Vilotte, J. P. (1998). The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures. Bull. Seismol. Soc. Am., 88(2), 368–392.

Komatitsch, D., and Tromp, J. (1999). Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int., 139(3), 806–822.

Komatitsch, D., and Tromp, J. (2002a). Spectral-element simulations of global seismic wave propagation—I. Validation. Geophys. J. Int., 149(2), 390–412.

Komatitsch, D., and Tromp, J. (2002b). Spectral-element simulations of global seismic wave propagation—Ⅱ. Three-dimensional models, oceans, rotation and self-gravitation. Geophys. J. Int., 150(1), 303–318.

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., Xie, Z. N., Bozdağ, E., de Andrade, E. S., Peter, D., Liu, Q. Y., and Tromp, J. (2016). Anelastic sensitivity kernels with parsimonious storage for adjoint tomography and full waveform inversion. Geophys. J. Int., 206(3), 1467–1478.

Lee, E. J., Chen, P., Jordan, T. H., Maechling, P. B., Denolle, M. A. M., and Beroza, G. C. (2014). Full-3-D tomography for crustal structure in southern California based on the scattering-integral and the adjoint-wavefield methods. J. Geophys. Res., 119(8), 6421–6451.

Liu, H. W., Ding, R. W., Liu, L., and Liu, H. (2013). Wavefield reconstruction methods for reverse time migration. J. Geophys. Eng., 10(1), 015004.

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

Liu, Q. Y., and Tromp, J. (2008). Finite-frequency sensitivity kernels for global seismic wave propagation based upon adjoint methods. Geophys. J. Int., 174(1), 265–286.

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., and Zhu, T. (2015). Source wavefield reconstruction using a linear combination of the boundary wavefield in reverse time migration. Geophysics, 80(6), S203–S212.

Liu, S. L., Yang D. H., Dong X. P., Liu Q. C., Zheng Y. C. (2017a). Element-by-element parallel spectral-element methods for 3-D teleseismic wave modeling. Solid Earth, 8(5), 969–986.

Liu, S. L., Yang D. H., and Ma J. (2017b). A modified symplectic PRK scheme for seismic wave modeling. Comput. Geosci., 99, 28–36.

Liu, X. J., Liu, Y. K., Huang, X. G, and Li, P. (2016). Least-squares reverse-time migration with cost-effective computation and memory storage. J. Appl. Geophys., 129, 200–208.

Liu, X. J., Liu, Y. K., Lu, H. Y., Hu, H., and Khan, M. (2017). Prestack correlative least-squares reverse time migration. Geophysics, 82(2), S159–S172.

Liu, Y. S., Teng, J. W., Lan, H. Q., Si, X., and Ma, X. Y. (2014). A comparative study of finite element and spectral element methods in seismic wavefield modeling. Geophysics, 79(2), T91–T104.

Liu, Y. S., Teng, J. W., Xu, T., Bai, Z. M., Lan, H. Q., and Badal, J. (2016). An efficient step-length formula for correlative least-squares reverse time migration. Geophysics, 81(4), S221–S238.

Liu, Y. S., Teng, J. W., Xu, T., and Badal, J. (2017a). Higher-order triangular spectral element method with optimized cubature points for seismic wavefield modeling. J. Computat. Phys., 336, 458–480.

Liu, Y. S., Teng, J. W., Xu, T., Wang, Y. H., Liu, Q. Y., and Badal, J. (2017b). Robust time-domain full waveform inversion with normalized zero-lag cross-correlation objective function. Geophys. J. Int., 209(1), 106–122.

Martin, R., Komatitsch, D., Gedney S. D., and Bruthiaux, E. (2010). 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), 17–42.

Newmark, N. M. (1959). A method of computation for structural dynamics. J. Eng. Mech. Div., 85(3), 67–94.

Nguyen, B. D., and McMechan, G. A. (2014). Five ways to avoid storing source wavefield snapshots in 2D elastic prestack reverse time migration. Geophysics, 80(1), S1–S18.

Peter, D., Komatitsch, D., Luo, Y., Martin, R., Le Goff, N., Casarotti, E., Le Loher, P., Magnoni, F., Liu, Q. Y., … Blitz, C. (2011). Forward and adjoint simulations of seismic wave propagation on fully unstructured hexahedral meshes. Geophys. J. Int., 186(2), 721–739.

Plessix, R. E. (2006). A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Int., 167(2), 495–503.

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

Pratt, R. G., and Shipp, R. M. (1999). Seismic waveform inversion in the frequency domain; part 2; fault delineation in sediments using crosshole data. Geophysics, 64(3), 902–914.

Shen, X. K., and Clapp, R. G. (2015). Random boundary condition for memory-efficient waveform inversion gradient computation. Geophysics, 80(6), R351–R359.

Shi, Y., and Wang, Y. H. (2016). Reverse time migration of 3D vertical seismic profile data. Geophysics, 81(1), S31–S38.

Sun, W. J., and Fu, L. Y. (2013). Two effective approaches to reduce data storage in reverse time migration. Comput. Geosci., 56, 69–75.

Symes, W. W. (2007). Reverse time migration with optimal checkpointing. Geophysics, 72(5), SM213–SM221.

Tan, S. R., and Huang, L. J. (2014). Reducing the computer memory requirement for 3D reverse-time migration with a boundary-wavefield extrapolation method. Geophysics, 79(5), S185–S194.

Tang, C., and Wang, D. (2012). Reverse time migration with source wavefield reconstruction in target imaging region. In Proceedings of the 74th EAGE Conference and Exhibition incorporating EUROPEC 2012 (pp. 3716-3720). Copenhagen, Denmark: SEG.222

Tang, Y. X. (2009). Target-oriented wave-equation least-squares migration/inversion with phase-encoded Hessian. Geophysics, 74(6), WCA95–WCA107.

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.

Tape, C., Liu, Q. Y., Maggi, A., and Tromp, J. (2010). Seismic tomography of the southern California crust based on spectral-element and adjoint methods. Geophys. J. Int., 180(1), 433–462.

Tarantola, A. (1984). Inversion of seismic reflection data in the acoustic approximation. Geophysics, 49(8), 1259–1266.

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

Tromp, J., Komatitsch, D., and Liu, Q. Y. (2008). Spectral-element and adjoint methods in seismology. Commun. Comput. Phys., 3(1), 1–32.

Vasmel, M., and Robertsson, J. O. A. (2016). Exact wavefield reconstruction on finite-difference grids with minimal memory requirements. Geophysics, 81(6), T303–T309.

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

Whitmore, N. D., and Lines, L. R. (1986). Vertical seismic profiling depth migration of a salt dome flank. Geophysics, 51(5), 1087–1109.

Wong, M., Biondi, B. L., and Ronen, S. (2015). Imaging with primaries and free-surface multiples by joint least-squares reverse time migration. Geophysics, 80(6), S223–S235.

Yang, P. L., Brossier, R., Métivier, L., and Virieux, J. (2016a). Wavefield reconstruction in attenuating media: a checkpointing-assisted reverse-forward simulation method. Geophysics, 81(6), R349–R362.

Yang, P. L., Brossier, R., and Virieux, J. (2016b). Wavefield reconstruction by interpolating significantly decimated boundaries. Geophysics, 81(5), T197–T209.

Zhu, H. J., Bozdağ, E., and Tromp, J. (2015). Seismic structure of the European upper mantle based on adjoint tomography. Geophys. J. Int., 201(1), 18–52.


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An efficient source wavefield reconstruction scheme using single boundary layer values for the spectral element method

YouShan Liu, Tao Xu, YangHua Wang, JiWen Teng, José Badal, HaiQiang Lan