none 10.1190/geo2020-0296.1 Society of Exploration Geophysicists Society of Exploration Geophysicists 186 123707803 5981 2025121212211584800 10.1190 2025-12-12T17:21:29Z 2021-01-28T18:20:43Z 8 Geophysics 0016-8033 1942-2156 05 01 2021 86 3 Na Fan Yangtze University 1 , Key Laboratory of Exploration Technology for Oil and Gas Resources of Ministry of Education, Wuhan 430100, China and , Wuhan, China. E-mail: fannachina@hotmail.com (corresponding author); tangxg@yangtzeu.edu.cn . Cooperative Innovation Center of Unconventional Oil and Gas (Ministry of Education & Hubei Province) 1 , Key Laboratory of Exploration Technology for Oil and Gas Resources of Ministry of Education, Wuhan 430100, China and , Wuhan, China. E-mail: fannachina@hotmail.com (corresponding author); tangxg@yangtzeu.edu.cn . http://orcid.org/0000-0003-0456-739X Xiao-Bi Xie University of California at Santa Cruz 2 , Institute of Geophysics and Planetary Physics, Santa Cruz, California 95064, USA. E-mail: xxie@ucsc.edu . Lian-Feng Zhao Chinese Academy of Sciences 3 , Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Beijing 100864, China. E-mail: zhaolf@mail.iggcas.ac.cn ; yaozx@mail.iggcas.ac.cn . http://orcid.org/0000-0002-1400-3064 Xin-Gong Tang Yangtze University 1 , Key Laboratory of Exploration Technology for Oil and Gas Resources of Ministry of Education, Wuhan 430100, China and , Wuhan, China. E-mail: fannachina@hotmail.com (corresponding author); tangxg@yangtzeu.edu.cn . Cooperative Innovation Center of Unconventional Oil and Gas (Ministry of Education & Hubei Province) 1 , Key Laboratory of Exploration Technology for Oil and Gas Resources of Ministry of Education, Wuhan 430100, China and , Wuhan, China. E-mail: fannachina@hotmail.com (corresponding author); tangxg@yangtzeu.edu.cn . Zhen-Xing Yao Chinese Academy of Sciences 3 , Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Beijing 100864, China. E-mail: zhaolf@mail.iggcas.ac.cn ; yaozx@mail.iggcas.ac.cn . ABSTRACT We have developed an optimal method to determine expansion parameters for flexible stencils in 2D scalar-wave finite-difference frequency-domain (FDFD) simulation. Our stencil only requires the involved grid points to be paired and rotationally symmetric around the central point. We apply this method to the transition zone in discontinuous-grid modeling, in which the key issue is designing particular FDFD stencils to correctly propagate the wavefield passing through the discontinuous interface. Our method can work in an FDFD discontinuous grid with arbitrary integer coarse- to fine-grid spacing ratios. Numerical examples are developed to determine how to apply this optimal method to discontinuous-grid FDFD schemes with spacing ratios of 3 and 5. The synthetic wavefields are highly consistent to those calculated using the conventional dense uniform grid, and the memory requirement and computational costs are greatly reduced. For velocity models with large contrasts, our discontinuous-grid FDFD method can significantly improve the computational efficiency in forward modeling, imaging, and full-waveform inversion. 03 19 2021 05 01 2021 01 28 2021 T143 T154 National Natural Science Foundation of China http://dx.doi.org/10.13039/501100001809 41604037 41630210 41674107 41874119 Cooperative Innovation Center of Unconventional Oil and Gas 501100023539 UOG2020-09 1 10.1190/crossmark-policy library.seg.org true 2020-05-22 2021-01-08 2021-01-08 2021-03-19 National Natural Science Foundation of China http://dx.doi.org/10.13039/501100001809 41604037 41630210 41674107 41874119 10.1190/geo2020-0296.1 https://pubs.geoscienceworld.org/geophysics/article/86/3/T143/595500/An-optimal-frequency-domain-finite-difference https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/86/3/T143/5380518/geo-2020-0296.1.pdf https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/86/3/T143/5380518/geo-2020-0296.1.pdf https://pubs.geoscienceworld.org/geophysics/article/86/3/t143/595500/an-optimal-frequency-domain-finite-difference 0037-1106 Bulletin of the Seismological Society of America Aoi 89 918 1999 10.1785/BSSA0890040918 3D finite-difference method using discontinuous grids Chinese Journal of Geophysics Cao 55 239 2012 A 17-point scheme and its numerical implementation for high-accuracy modeling of frequency-domain acoustic equation 0016-8033 Geophysics Chen 77 6 T201 2012 10.1190/geo2011-0389.1 An average-derivative optimal scheme for frequency-domain scalar wave equation 0016-8033 Geophysics Chen 81 6 T339 2016 10.1190/geo2016-0041.1 Modeling of frequency-domain elastic-wave equation with an average-derivative optimal method 0016-8033 Geophysics Chen 83 4 T209 2018 10.1190/geo2017-0641.1 An average-derivative optimal scheme for modeling of the frequency-domain 3D elastic wave equation 0926-9851 Journal of Applied Geophysics Chu 76 44 2012 10.1016/j.jappgeo.2011.09.027 Nonuniform grid implicit spatial finite difference method for acoustic wave modeling in tilted transversely isotropic media Etgen 33 2007 A tutorial on optimizing time domain finite-difference schemes 0033-4553 Pure and Applied Geophysics Falk 148 77 1996 10.1007/BF00882055 Tube wave modeling by the finite-difference method with varying grid spacing Chinese Journal of Geophysics Fan 61 1095 2018 An optimal method for frequency-domain finite-difference solution of 3D scalar wave equation 0016-8033 Geophysics Fan 80 4 T175 2015 10.1190/geo2015-0001.1 A discontinuous collocated-grid implementation for high-order finite-difference modeling 0016-8033 Geophysics Fan 82 3 T121 2017 10.1190/geo2016-0457.1 A general optimal method for 2D frequency-domain finite-difference solution of scalar wave equation 0016-8033 Geophysics Fan 83 4 T235 2018 10.1190/geo2017-0535.1 A discontinuous-grid finite-difference scheme for frequency-domain 2D scalar wave modeling Exploration Geophysics Gu 44 156 2013 10.1071/EG12064 A 21-point finite difference scheme for 2D frequency-domain elastic wave modelling 0016-8025 Geophysical Prospecting Holberg 35 629 1987 10.1111/j.1365-2478.1987.tb00841.x Computational aspects of the choice of operator and sampling interval for numerical differentiation in large-scale simulation of wave phenomena Chinese Journal of Geophysics Huang 52 176 2009 High-order finite-difference method in seismic wave simulation with variable grids and local time-steps Chinese Journal of Geophysics Huang 52 2870 2009 Staggered-grid high-order finite-difference method in elastic wave simulation with variable grids and local time-steps 0956-540X Geophysical Journal International Hustedt 157 1269 2004 10.1111/j.1365-246X.2004.02289.x Mixed-grid and staggered-grid finite-difference methods for frequency-domain acoustic wave modelling 0016-8025 Geophysical Prospecting Jastram 40 157 1992 10.1111/j.1365-2478.1992.tb00369.x Acoustic modelling on a grid of vertically varying spacing 0016-8025 Geophysical Prospecting Jastram 42 357 1994 10.1111/j.1365-2478.1994.tb00215.x Elastic modelling on a grid with vertically varying spacing 0016-8033 Geophysics Jo 61 529 1996 10.1190/1.1443979 An optimal 9-point, finite-difference, frequency-space, 2-D scalar wave extrapolator 0037-1106 Bulletin of the Seismological Society of America Kang 94 207 2004 10.1785/0120030080 Finite-difference seismic simulation combining discontinuous grids with locally variable timesteps 0956-540X Geophysical Journal International Kristek 183 1401 2010 10.1111/j.1365-246X.2010.04775.x Stable discontinuous staggered grid in the finite-difference modelling of seismic motion 0926-9851 Journal of Applied Geophysics Li 159 1 2018 10.1016/j.jappgeo.2018.07.014 Modeling of frequency-domain elastic-wave equation with a general optimal scheme 0016-8033 Geophysics Li 83 5 T265 2018 10.1190/geo2017-0132.1 A generalized average-derivative optimal finite-difference scheme for 2D frequency-domain acoustic-wave modeling on continuous nonuniform grids Earthquake Science Liu 27 169 2014 10.1007/s11589-013-0055-7 Finite-difference modeling with variable grid-size and adaptive time-step in porous media 0016-8033 Geophysics Min 65 884 2000 10.1190/1.1444785 Improved frequency-domain elastic wave modeling using weighted-averaging difference operators 0956-540X Geophysical Journal International Moczo 99 321 1989 10.1111/j.1365-246X.1989.tb01691.x Finite-difference technique for SH-waves in 2-D media using irregular grids — Application to the seismic response problem 10.1190/segam2015-5931765.1 Nie S. Wang Y. Olsen K. Day S. , 2015, Stable discontinuous staggered finite difference method for elastic wave simulations: 85th Annual International Meeting, SEG, Expanded Abstracts, 3774–3778, doi: http://dx.doi.org/10.1190/segam2015-5931765.1. 0016-8033 Geophysics Oliveira 68 672 2003 10.1190/1.1567237 A fourth-order finite-difference method for the acoustic wave equation on irregular grids 0016-8033 Geophysics Operto 79 5 T257 2014 10.1190/geo2013-0478.1 Computationally efficient three-dimensional acoustic finite-difference frequency-domain seismic modeling in vertical transversely isotropic media with sparse direct solver 0016-8033 Geophysics Operto 72 5 SM195 2007 10.1190/1.2759835 3D finite-difference frequency-domain modeling of visco-acoustic wave propagation using a massively parallel direct solver: A feasibility study 0016-8033 Geophysics Operto 74 5 T75 2009 10.1190/1.3157243 Finite-difference frequency-domain modeling of viscoacoustic wave propagation in 2D tilted transversely isotropic (TTI) media 0016-8033 Geophysics Opršal 64 240 1999 10.1190/1.1444520 Elastic finite-difference method for irregular grids 0037-1106 Bulletin of the Seismological Society of America Pitarka 89 54 1999 10.1785/BSSA0890010054 3D elastic finite-difference modeling of seismic motion using staggered grids with nonuniform spacing 0016-8033 Geophysics Plessix 74 6 WCC149 2009 10.1190/1.3211198 Three-dimensional frequency-domain full-waveform inversion with an iterative solver 0016-8033 Geophysics Shin 63 289 1998 10.1190/1.1444323 A frequency-space 2-D scalar wave extrapolator using extended 25-point finite-difference operator 0016-8033 Geophysics Štekl 63 1779 1998 10.1190/1.1444472 Accurate viscoelastic modeling by frequency-domain finite differences using rotated operators 0016-8033 Geophysics Tang 80 6 T211 2015 10.1190/geo2014-0124.1 An adaptable 17-point scheme for high-accuracy frequency-domain acoustic wave modeling in 2D constant density media 10.1190/1.3059269 Vigh D. Starr E. W. , 2008, Comparisons for waveform inversion, time domain or frequency domain?: 78th Annual International Meeting, SEG, Expanded Abstracts, 1890–1894, doi: http://dx.doi.org/10.1190/1.3059269. 0016-8033 Geophysics Virieux 74 6 WCC1 2009 10.1190/1.3238367 An overview of full-waveform inversion in exploration geophysics Pure Applied Geophysics Wang 176 5391 2019 10.1007/s00024-019-02277-2 Temporal high-order time–space domain finite-difference methods for modeling 3D acoustic wave equations on general cuboid grids 0016-8033 Geophysics Wang 84 5 T335 2019 10.1190/geo2018-0732.1 Nearly perfectly matched layer absorber for viscoelastic wave equations 0037-1106 Bulletin of the Seismological Society of America Wang 91 1741 2001 10.1785/0120000236 Viscoelastic wave simulation in basins by a variable-grid finite-difference method Petroleum Science Wang 16 1001 2019 10.1007/s12182-019-00368-2 3D variable-grid full-waveform inversion on GPU 0956-540X Geophysical Journal International Yang 208 148 2016 10.1093/gji/ggw380 Simulation of seismic wave propagation in 2-D poroelastic media using weighted-averaging finite difference stencils in the frequency–space domain 10.1190/segam2015-5871009.1 Zhang H. Zhang B. Liu B. Liu H. Shi X. , 2015, Frequency-space domain high-order modeling based on an average-derivative optimal method: 85th Annual International Meeting, SEG, Expanded Abstracts, 3749–3753, doi: http://dx.doi.org/10.1190/segam2015-5871009.1. 0956-540X Geophysical Journal International Zhang 192 1179 2013 10.1093/gji/ggs069 Stable discontinuous grid implementation for collocated-grid finite-difference seismic wave modelling