An implicit finite-difference operator for the Helmholtz equation

Handle URI:
http://hdl.handle.net/10754/597527
Title:
An implicit finite-difference operator for the Helmholtz equation
Authors:
Chu, Chunlei; Stoffa, Paul L.
Abstract:
We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly. © 2012 Society of Exploration Geophysicists.
Citation:
OCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html><head> <title>Current Links for doi: 10.1190/geo2011-0314.1 </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <!--geophysics.html--></head><style> body { font-size: 100%; font-family: arial, times; } #prime { padding: 5px; } #second { padding: 5px; } #links { padding: 5px; } #data { padding:5px; } #top { padding:5px; }</style><body><div id="top"> <img src="/docs/iPage/images/SEG_corp_logo.jpg" alt="SEG Corp Logo"/> <p> The Society of Exploration Geophysicists is a not-for-profit organization that promotes the science of applied geophysics and the education of geophysicists. SEG, founded in 1930, fosters the expert and ethical practice of geophysics in the exploration and development of natural resources, in characterizing the near surface, and in mitigating earth hazards. The Society, which has more than 33 000 members in 138 countries, fulfills its mission through its publications, conferences, forums, Web sites, and educational opportunities. </p> <hr/></div><div id="data"> <table width="450"> <tr><td><b>An implicit finite-difference operator for the Helmholtz equation</b></td></tr><tr><td>Chu,Chunlei et al.</td></tr><tr><td><i>GEOPHYSICS</i>(2012),77(4):T97</td></tr><tr><td><a href='http://dx.doi.org/10.1190/geo2011-0314.1'>http://dx.doi.org/10.1190/geo2011-0314.1</td></tr> </table> <hr/></div><div id="links"> <p> This article is available from multiple sources. <br/>Please click on the logo of the service to which you have a subscription, or click any logo to obtain pay-per-view access. </p> <div id="prime"> <a href=http://library.seg.org/doi/abs/10.1190/geo2011-0314.1><img src="/docs/iPage/images/segdl_logo.gif" alt="SEGDL Logo"/></a> </div> <div id="second"> <a href=http://geophysics.geoscienceworld.org/cgi/doi/10.1190/geo2011-0314.1><img src="/docs/iPage/images/gswbanner.gif" alt="GSW Logo"/></a> </div></div></body></html>
Publisher:
Society of Exploration Geophysicists
Journal:
GEOPHYSICS
Issue Date:
Jul-2012
DOI:
10.1190/GEO2011-0314.1
Type:
Article
ISSN:
0016-8033; 1942-2156
Sponsors:
Paul Stoffa would like to acknowledge the King Abdullah University of Science and Technology (KAUST) for their support of his research. We are grateful for the constructive comments from the reviewers and associate editor Jeffrey Shragge, which helped improve the original manuscript significantly. We thank ConocoPhillips for permission to publish this work.
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Publications Acknowledging KAUST Support

Full metadata record

DC FieldValue Language
dc.contributor.authorChu, Chunleien
dc.contributor.authorStoffa, Paul L.en
dc.date.accessioned2016-02-25T12:41:28Zen
dc.date.available2016-02-25T12:41:28Zen
dc.date.issued2012-07en
dc.identifier.citationOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"><html><head> <title>Current Links for doi: 10.1190/geo2011-0314.1 </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <!--geophysics.html--></head><style> body { font-size: 100%; font-family: arial, times; } #prime { padding: 5px; } #second { padding: 5px; } #links { padding: 5px; } #data { padding:5px; } #top { padding:5px; }</style><body><div id="top"> <img src="/docs/iPage/images/SEG_corp_logo.jpg" alt="SEG Corp Logo"/> <p> The Society of Exploration Geophysicists is a not-for-profit organization that promotes the science of applied geophysics and the education of geophysicists. SEG, founded in 1930, fosters the expert and ethical practice of geophysics in the exploration and development of natural resources, in characterizing the near surface, and in mitigating earth hazards. The Society, which has more than 33 000 members in 138 countries, fulfills its mission through its publications, conferences, forums, Web sites, and educational opportunities. </p> <hr/></div><div id="data"> <table width="450"> <tr><td><b>An implicit finite-difference operator for the Helmholtz equation</b></td></tr><tr><td>Chu,Chunlei et al.</td></tr><tr><td><i>GEOPHYSICS</i>(2012),77(4):T97</td></tr><tr><td><a href='http://dx.doi.org/10.1190/geo2011-0314.1'>http://dx.doi.org/10.1190/geo2011-0314.1</td></tr> </table> <hr/></div><div id="links"> <p> This article is available from multiple sources. <br/>Please click on the logo of the service to which you have a subscription, or click any logo to obtain pay-per-view access. </p> <div id="prime"> <a href=http://library.seg.org/doi/abs/10.1190/geo2011-0314.1><img src="/docs/iPage/images/segdl_logo.gif" alt="SEGDL Logo"/></a> </div> <div id="second"> <a href=http://geophysics.geoscienceworld.org/cgi/doi/10.1190/geo2011-0314.1><img src="/docs/iPage/images/gswbanner.gif" alt="GSW Logo"/></a> </div></div></body></html>en
dc.identifier.issn0016-8033en
dc.identifier.issn1942-2156en
dc.identifier.doi10.1190/GEO2011-0314.1en
dc.identifier.urihttp://hdl.handle.net/10754/597527en
dc.description.abstractWe have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly. © 2012 Society of Exploration Geophysicists.en
dc.description.sponsorshipPaul Stoffa would like to acknowledge the King Abdullah University of Science and Technology (KAUST) for their support of his research. We are grateful for the constructive comments from the reviewers and associate editor Jeffrey Shragge, which helped improve the original manuscript significantly. We thank ConocoPhillips for permission to publish this work.en
dc.publisherSociety of Exploration Geophysicistsen
dc.subjectAcousticen
dc.subjectFinite differenceen
dc.subjectModelingen
dc.titleAn implicit finite-difference operator for the Helmholtz equationen
dc.typeArticleen
dc.identifier.journalGEOPHYSICSen
dc.contributor.institutionConocoPhillips, Houston, United Statesen
dc.contributor.institutionUniversity of Texas at Austin, Austin, United Statesen
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