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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.<\/jats:p>","DOI":"10.1190\/geo2020-0296.1","type":"journal-article","created":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T18:20:43Z","timestamp":1611858043000},"page":"T143-T154","update-policy":"https:\/\/doi.org\/10.1190\/crossmark-policy","source":"Crossref","is-referenced-by-count":8,"title":["An optimal frequency-domain finite-difference operator with a flexible stencil and its application in discontinuous-grid modeling"],"prefix":"10.1190","volume":"86","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0456-739X","authenticated-orcid":false,"given":"Na","family":"Fan","sequence":"first","affiliation":[{"name":"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 ."},{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiao-Bi","family":"Xie","sequence":"additional","affiliation":[{"name":"University of California at Santa Cruz 2 , Institute of Geophysics and Planetary Physics, Santa Cruz, California 95064, USA. E-mail: xxie@ucsc.edu ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1400-3064","authenticated-orcid":false,"given":"Lian-Feng","family":"Zhao","sequence":"additional","affiliation":[{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin-Gong","family":"Tang","sequence":"additional","affiliation":[{"name":"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 ."},{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhen-Xing","family":"Yao","sequence":"additional","affiliation":[{"name":"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 ."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"186","published-online":{"date-parts":[[2021,3,19]]},"reference":[{"key":"2025121212211584800_r1","doi-asserted-by":"crossref","first-page":"918","DOI":"10.1785\/BSSA0890040918","article-title":"3D finite-difference method using discontinuous grids","volume":"89","author":"Aoi","year":"1999","journal-title":"Bulletin of the Seismological Society of America","ISSN":"https:\/\/id.crossref.org\/issn\/0037-1106","issn-type":"print"},{"key":"2025121212211584800_r2","first-page":"239","article-title":"A 17-point scheme and its numerical implementation for high-accuracy modeling of frequency-domain acoustic equation","volume":"55","author":"Cao","year":"2012","journal-title":"Chinese Journal of Geophysics"},{"issue":"6","key":"2025121212211584800_r3","doi-asserted-by":"crossref","first-page":"T201","DOI":"10.1190\/geo2011-0389.1","article-title":"An average-derivative 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