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Third-order symplectic integration method with inverse time dispersion transform for long-term simulation. (English) Zbl 1349.65514

Summary: The symplectic integration method is popular in high-accuracy numerical simulations when discretizing temporal derivatives; however, it still suffers from time-dispersion error when the temporal interval is coarse, especially for long-term simulations and large-scale models. We employ the inverse time dispersion transform (ITDT) to the third-order symplectic integration method to reduce the time-dispersion error. First, we adopt the pseudospectral algorithm for the spatial discretization and the third-order symplectic integration method for the temporal discretization. Then, we apply the ITDT to eliminate time-dispersion error from the synthetic data. As a post-processing method, the ITDT can be easily cascaded in traditional numerical simulations. We implement the ITDT in one typical exiting third-order symplectic scheme and compare its performances with the performances of the conventional second-order scheme and the rapid expansion method. Theoretical analyses and numerical experiments show that the ITDT can significantly reduce the time-dispersion error, especially for long travel times. The implementation of the ITDT requires some additional computations on correcting the time-dispersion error, but it allows us to use the maximum temporal interval under stability conditions; thus, its final computational efficiency would be higher than that of the traditional symplectic integration method for long-term simulations. With the aid of the ITDT, we can obtain much more accurate simulation results but with a lower computational cost.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65P10 Numerical methods for Hamiltonian systems including symplectic integrators

Software:

SHASTA
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Full Text: DOI

References:

[1] Carcione, J. M.; Herman, G. C.; Ten Kroode, A., Seismic modeling, Geophysics, 67, 1304-1325 (2002)
[2] Alterman, Z.; Karal, F., Propagation of elastic waves in layered media by finite difference methods, Bull. Seismol. Soc. Am., 58, 367-398 (1968)
[3] Alford, R.; Kelly, K.; Boore, D. M., Accuracy of finite-difference modeling of the acoustic wave equation, Geophysics, 39, 834-842 (1974)
[4] Virieux, J., SH-wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 49, 1933-1942 (1984)
[5] Gazdag, J., Modeling of the acoustic wave equation with transform methods, Geophysics, 46, 854-859 (1981)
[6] Kosloff, D. D.; Baysal, E., Forward modeling by a Fourier method, Geophysics, 47, 1402-1412 (1982)
[7] Kosloff, D. D.; Reshef, M.; Loewenthal, D., Elastic wave calculations by the Fourier method, Bull. Seismol. Soc. Am., 74, 875-891 (1984)
[8] Fornberg, B., The pseudospectral method: comparisons with finite differences for the elastic wave equation, Geophysics, 52, 483-501 (1987)
[9] Marfurt, K. J., Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations, Geophysics, 49, 533-549 (1984)
[10] Komatitsch, D.; Vilotte, J.-P., The spectral element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures, Bull. Seismol. Soc. Am., 88, 368-392 (1998) · Zbl 0974.74583
[11] Dormy, E.; Tarantola, A., Numerical simulation of elastic wave propagation using a finite volume method, J. Geophys. Res., Solid Earth (1978-2012), 100, 2123-2133 (1995)
[12] Dablain, M. A., The application of high-order differencing to the scalar wave equation, Geophysics, 51, 54-66 (1986)
[13] Etgen, J. T., High-order finite-difference reverse time migration with the 2-way non-reflecting wave equation, 133-146 (1986), Stanford Exploration Project, Report-48
[14] Etgen, J. T., Evaluating finite-difference operators applied to wave simulation, 243-258 (1988), Stanford Exploration Project, Report-57
[15] Liu, Y.; Sen, M. K., A new time-space domain high-order finite-difference method for the acoustic wave equation, J. Comput. Phys., 228, 8779-8806 (2009) · Zbl 1176.65093
[16] Liu, Y.; Sen, M. K., Time-space domain dispersion-relation-based finite-difference method with arbitrary even-order accuracy for the 2D acoustic wave equation, J. Comput. Phys., 232, 327-345 (2013)
[17] Holberg, O., Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena, Geophys. Prospect., 35, 629-655 (1987)
[18] Etgen, J. T., A tutorial on optimizing time domain finite-difference schemes: “Beyond Holberg”, 33-43 (2007), Stanford Exploration Project, Report-129
[19] Zhou, H.; Zhang, G., Prefactored optimized compact finite-difference schemes for second spatial derivatives, Geophysics, 76, WB87-WB95 (2011)
[20] Chu, C.; Stoffa, P. L., Determination of finite-difference weights using scaled binomial windows, Geophysics, 77, W17-W26 (2012)
[21] Zhang, J.; Yao, Z., Optimized explicit finite-difference schemes for spatial derivatives using maximum norm, J. Comput. Phys., 250, 511-526 (2013) · Zbl 1349.65358
[22] Zhang, J.; Yao, Z., Optimized finite-difference operator for broadband seismic wave modeling, Geophysics, 78, A13-A18 (2013)
[23] Liu, Y., Globally optimal finite-difference schemes based on least squares, Geophysics, 78, T113-T132 (2013)
[24] Tan, S.; Huang, L., A staggered-grid finite-difference scheme optimized in the time-space domain for modeling scalar-wave propagation in geophysical problems, J. Comput. Phys., 276, 613-634 (2014) · Zbl 1349.74366
[25] Wang, Y.; Liang, W.; Nashed, Z.; Li, X.; Liang, G.; Yang, C., Seismic modeling by optimizing regularized staggered-grid finite-difference operators using a time-space-domain dispersion-relationship-preserving method, Geophysics, 79, T277-T285 (2014)
[26] Sun, W.; Zhou, B.; Fu, L. Y., A staggered-grid convolutional differentiator for elastic wave modelling, J. Comput. Phys., 301, 59-76 (2015) · Zbl 1349.74365
[27] Boris, J. P.; Book, D. L., Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 38-69 (1973) · Zbl 0251.76004
[28] Fei, T.; Larner, K., Elimination of numerical dispersion in finite-difference modeling and migration by flux-corrected transport, Geophysics, 60, 1830-1842 (1995)
[29] Yang, D.; Liu, E.; Zhang, Z.; Teng, J., Finite-difference modelling in two-dimensional anisotropic media using a flux-corrected transport technique, Geophys. J. Int., 148, 320-328 (2002)
[30] Yang, D.; Teng, J.; Zhang, Z.; Liu, E., A nearly analytic discrete method for acoustic and elastic wave equations in anisotropic media, Bull. Seismol. Soc. Am., 93, 882-890 (2003)
[31] Yang, D.; Lu, M.; Wu, R.; Peng, J., An optimal nearly analytic discrete method for 2D acoustic and elastic wave equations, Bull. Seismol. Soc. Am., 94, 1982-1991 (2004)
[32] Tong, P.; Yang, D.; Hua, B.; Wang, M., A high-order stereo-modeling method for solving wave equations, Bull. Seismol. Soc. Am., 103, 811-833 (2013)
[33] Stoffa, P. L.; Pestana, R. C., Numerical solution of the acoustic wave equation by the rapid expansion method (REM) - a one step time evolution algorithm, (79th Annual International Meeting, SEG, Expanded Abstracts (2009), Society of Exploration Geophysicists), 2672-2676
[34] Chen, J. B., High-order time discretizations in seismic modeling, Geophysics, 72, SM115-SM122 (2007)
[35] Zhang, Y.; Zhang, G.; Yingst, D.; Sun, J., Explicit marching method for reverse time migration, (77th Annual International Meeting, SEG, Expanded Abstracts (2007), Society of Exploration Geophysicists), 2300-2304
[36] Soubaras, R.; Zhang, Y., Two-step explicit marching method for reverse time migration, (78th Annual International Meeting, SEG, Expanded Abstracts (2008), Society of Exploration Geophysicists)
[37] Zhang, Y.; Zhang, G., One-step extrapolation method for reverse time migration, Geophysics, 74, A29-A33 (2009)
[38] Tal-Ezer, H.; Kosloff, D.; Koren, Z., An accurate scheme for seismic forward modelling, Geophys. Prospect., 35, 479-490 (1987)
[39] Kosloff, D.; Queiroz Filho, A.; Tessmer, E.; Behle, A., Numerical solution of the acoustic and elastic wave equations by a new rapid expansion method, Geophys. Prospect., 37, 383-394 (1989)
[40] Pestana, R. C.; Stoffa, P. L., Time evolution of the wave equation using rapid expansion method, Geophysics, 75, T121-T131 (2010)
[41] Tessmer, E., Using the rapid expansion method for accurate time-stepping in modeling and reverse-time migration, Geophysics, 76, S177-S185 (2011)
[42] Fomel, S.; Ying, L.; Song, X., Seismic wave extrapolation using lowrank symbol approximation, (80th Annual International Meeting, SEG, Expanded Abstracts (2010), Society of Exploration Geophysicists)
[43] Fomel, S.; Ying, L.; Song, X., Seismic wave extrapolation using lowrank symbol approximation, Geophys. Prospect., 61, 526-536 (2012)
[44] Song, X.; Fomel, S., Fourier finite-difference wave propagation, Geophysics, 76, T123-T129 (2011)
[45] Song, X.; Nihei, K.; Stefani, J., Seismic modeling in acoustic variable-density media by Fourier finite differences, (82th Annual International Meeting, SEG, Expanded Abstracts (2012), Society of Exploration Geophysicists)
[46] Song, X.; Fomel, S.; Ying, L., Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation, Geophys. J. Int., 193, 960-969 (2013)
[47] Li, Y. E.; Wong, M.; Clapp, R., Equivalent accuracy at a fraction of the cost: overcoming temporal dispersion (2013), Stanford Exploration Project, Report-150
[48] Stork, C., Eliminating nearly all dispersion error from FD modeling and RTM with minimal cost increase, (75th Annual International Conference and Exhibition, EAGE, Extended Abstracts, Tu 11 07 (2013))
[49] Liu, H.; Dai, N.; Niu, F.; Wu, W., An explicit time evolution method for acoustic wave propagation, Geophysics, 79, T117-T124 (2014)
[50] Dai, N.; Liu, H.; Wu, W., Solutions to numerical dispersion error of time FD in RTM, (84th Annual International Meeting, SEG, Expanded Abstracts (2014), Society of Exploration Geophysicists), 4027-4031
[51] Wang, M.; Xu, S., Finite-difference time dispersion transforms for wave propagation, Geophysics, 80, WD19-WD25 (2015)
[52] Wang, M.; Xu, S., Time dispersion prediction and correction for wave propagation, (85th Annual International Meeting, SEG, Expanded Abstracts (2015), Society of Exploration Geophysicists), 3677-3681
[53] Chen, J. B., Lax-Wendroff and Nyström methods for seismic modelling, Geophys. Prospect., 57, 931-941 (2009)
[54] Nyström, E. J., Über die numerische Integration von Differentialgleichungen (1925), Societas scientiarum Fennica, (Mitgeteilt am 23 Sept. 1925 von E. Lindelöf und K.F. Sundman) · JFM 51.0427.01
[55] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems (1993), Springer: Springer Berlin · Zbl 0789.65048
[56] Chen, J. B., Modeling the scalar wave equation with Nyström methods, Geophysics, 71, T151-T158 (2006)
[57] Ruth, R. D., A canonical integration technique, IEEE Trans. Nucl. Sci., 30, 2669-2671 (1983)
[58] Yoshida, H., Construction of higher order symplectic integrators, Phys. Lett. A, 150, 262-268 (1990)
[59] Qin, M. Z.; Zhang, M. Q., Multi-stage symplectic schemes of two kinds of Hamiltonian systems for wave equations, Comput. Math. Appl., 19, 51-62 (1990) · Zbl 0695.65072
[60] Chen, J. B.; Liu, H., Optimization approximation with separable variables for the one-way wave operator, Geophys. Res. Lett., 31, Article L06613 pp. (2004)
[61] Fang, G.; Fomel, S.; Du, Q.; Hu, J., Lowrank seismic-wave extrapolation on a staggered grid, Geophysics, 79, T157-T168 (2014)
[62] Etgen, J. T.; Brandsberg-Dahl, S., The pseudo-analytical method: application of pseudo-Laplacians to acoustic and acoustic anisotropic wave propagation, (79th Annual International Meeting, SEG, Expanded Abstracts (2009), Society of Exploration Geophysicists), 2552-2556
[63] Li, X.; Wang, W.; Lu, M.; Zhang, M.; Li, Y., Structure-preserving modelling of elastic waves: a symplectic discrete singular convolution differentiator method, Geophys. J. Int., 188, 1382-1392 (2012)
[64] Komatitsch, D.; Tromp, J., Spectral-element simulations of global seismic wave propagation - II. Three-dimensional models, oceans, rotation and self-gravitation, Geophys. J. Int., 150, 303-318 (2002)
[65] Liu, S.; Li, X.; Wang, W.; Liu, Y.; Zhang, M.; Zhang, H., A new kind of optimal second-order symplectic scheme for seismic wave simulations, Sci. China Earth Sci., 44, 283-291 (2014), (in Chinese)
[66] Feng, K.; Qin, M. Z., Symplectic Geometric Algorithms for Hamiltonian Systems (2003), Zhejiang Science & Technology Press: Zhejiang Science & Technology Press Hangzhou, China, (in Chinese)
[67] Ma, X.; Yang, D.; Liu, F., A nearly analytic symplectically partitioned Runge-Kutta method for 2-D seismic wave equations, Geophys. J. Int., 187, 480-496 (2011)
[68] McLachlan, R. I.; Atela, P., The accuracy of symplectic integrators, Nonlinearity, 5, 541-562 (1992) · Zbl 0747.58032
[69] Wang, W.; Li, X., A new solution to the third-order non-gradient symplectic integration algorithm, Wuhan Univ. J. Nat. Sci., 58, 221-228 (2012), (in Chinese) · Zbl 1265.65262
[70] Iwatsu, R., Two new solutions to the third-order symplectic integration method, Phys. Lett. A, 373, 3056-3060 (2009) · Zbl 1233.70005
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