### Numerical modeling of elastic wave in frequency-domain by using staggered grid fourth-order finite-difference scheme

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Simulation of elastic wave propagation is an important method for oil and gas exploration. Accuracy and efficiency of elastic wave simulation in complex geological environment are always the focus issue. In order to improve the accuracy and efficient in numerical modeling of elastic modeling, a staggered grid fourth-order finite-difference scheme of modeling elastic wave in frequency-domain is developed, which can provide stable numerical solution with fewer number of grid points per wavelength. The method is implemented on first-order velocity-stress equation and a parsimonious spatial staggered-grid with fourth-order approximation of the first-order derivative operator. Numerical tests show that the accuracy of the fourth-order staggered-grid stencil is superior to that of the mixed-grid and other conventional finite difference stencils, especially in terms of shear-wave phase velocity. Measures of mass averaging acceleration and optimization of finite difference coefficients are taken to improve the accuracy of numerical results. Meanwhile, the numerical accuracy of the finite difference scheme can be further improved by enlarging the mass averaging area at the price of expanding the bandwidth of the impedance matrix that results in the reduction of the number of grid points to 3 per shear wavelength and computer storage requirement in simulation of practical models. In our scheme, the phase velocities of compressional and shear wave are insensitive to Poisson's ratio that does not occur to conventional finite difference scheme in most cases, and also the elastic wave modeling can degenerate to acoustic case automatically when the medium is pure fluid or gas. Furthermore, the staggered grid scheme developed in this study is suitable for modeling waves propagating in media with coupling fluid-solid interfaces that are not resolved very well for previous finite difference method.

**Cited as**: Ma, C., Gao, Y., Lu, C. Numerical modeling of elastic wave in frequency-domain by using staggered grid fourth-order finite-difference scheme. Advances in Geo-Energy Research, 2019, 3(4): 410-423, doi: 10.26804/ager.2019.04.08

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Arntsen, B., Nebel, A., Amundsen, L. Visco-acoustic ﬁnite difference modeling in the frequency domain. J. Seism. Explor. 1998, 7(1): 45-64.

B ´erenger, J.P. A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 1994, 114(2): 185-200.

Chillara, V.K., Ren, B., Lissenden, C.J. Guided wave mode selection for inhomogeneous elastic waveguides using frequency domain ﬁnite element approach. Ultrasonics 2016, 67: 199-211.

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

Erlangga, Y., Oosterlee, C., Vuik, C. A novel multigrid based preconditioner for heterogeneous Helmholtz problem. SIAM J. Sci. Comput. 2006, 27(4): 1471-1492.

Gelis, C., Virieux, J., Grandiean, G. Two-dimensional elastic full waveform inversion using Born and Rytov formulations in the frequency domain. Geophys. J. Int. 2007, 168(2): 605-633.

Hustedt, B., Operto, S., Virieux, J. Mixed-grid and staggered-grid ﬁnite-difference methods for frequency-domain acoustic wave modelling. Geophys. J. Int. 2004, 157(3): 1269-1296.

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

Levander, A.R. Fourth-order ﬁnite-difference P-SV seismograms. Geophysics 1988, 53(11): 1425-1436.

Lines, L.R., Treitel, S. Tutorial: A review of least-squares inversion and its application to geophysical problems. Geophys. Prospect. 1984, 32(2): 159-186.

Luo, Y., Schuster, G. Parsimonious staggered grid ﬁnite differencing of the wave equation. Geophys. Res. Lett. 1990, 17(2), 155-158.

Ma, C., Shen, J.S. 2D elastic ﬁnite-difference modeling of seismic response in fractured anisotropic media. Presented at the 75th EAGE conference & Exhibition incorporating SPE EUROPEC 2013, June, 2013.

Madariaga, R. Dynamics of an expanding circular fault. B. Seismol. Soc. Am. 1976, 66(3): 639-666.

Matuszyk, P.J., Demkowicz, L.F., Torres-Verdin, C. Solution of coupled acoustic-elastic wave propagation problems with anelastic attenuation using automatic hp-adaptivity. Comput. Methods Appl. Mech. Eng. 2012, 213-216: 299-313.

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

Operto, S., Virieux, J.P., Amestoy, J.L., et al. 3D ﬁnite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: a feasibility study. Geophysics 2007, 72(5): 195-211.

Pan, G., Abubakar, A., Habashy, T.M. An effective perfectly matched layer design for acoustic fourth-order frequency-domain ﬁnite-difference scheme. Geophys. J. Int. 2012, 188(1): 211-222.

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

Plessix, R.E. Three-dimensional frequency-domain full-waveform inversion with an iterative solver. Geophysics 2009, 74(6): 149-157.

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

Pratt, R.G. Inverse theory applied to multi-source cross-hole tomography, part 2: Elastic wave-equation method. Geophys. Prospect. 1990b, 38(3): 311-329.

Saenger, E.H., Gold, N., Shapiro, A. Modeling the propagation of elastic waves using a modiﬁed ﬁnite-difference grid. Wave Motion 2000, 31(1): 77-92.

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

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

Takekawa, J., Mikada, H. A mesh-free ﬁnite-difference method for elastic wave propagation in the frequency-domain. Comput. Geosci. 2018, 118: 65-78.

Virieux, J. SH wave propagation in heterogeneous media, velocity stress ﬁnite difference method. Geophysics 1984, 49(11): 1933-1942.

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

Wang, Y., Liang, W. Optimized ﬁnite difference methods for seismic acoustic wave modeling, in Computational and Experimental Studies of Acoustic Waves, edited by M. Reyhanoglu, IntechOpen, London, pp. 3-26, 2018.

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