{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:31:36Z","timestamp":1787337096526,"version":"build-2736575974"},"reference-count":54,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2006,1]]},"abstract":"<jats:p>We propose a local level set method for constructing the geometrical optics term in the paraxial formulation for the high frequency asymptotics of two-dimensional (2-D) acoustic wave equations. The geometrical optics term consists of two multivalued functions: a travel-time function satisfying the eikonal equation locally and an amplitude function solving a transport equation locally. The multivalued travel-times are obtained by solving a level set equation and a travel-time equation with a forcing term. The multivalued amplitudes are computed by a newEulerian formula based on the gradients of travel-times and takeoff angles. As a byproduct the method is also able to capture the caustic locations. The proposed Eulerian method has complexity of $O(N^2{\\rm Log }N)$, rather than $O(N^4)$ as typically seen in the Lagrangian ray tracing method. Several examples including the well-known Marmousi synthetic model illustrate the accuracy and efficiency of the Eulerian method.<\/jats:p>","DOI":"10.1137\/030601673","type":"journal-article","created":{"date-parts":[[2006,3,15]],"date-time":"2006-03-15T21:00:21Z","timestamp":1142456421000},"page":"206-223","source":"Crossref","is-referenced-by-count":21,"title":["A Local Level Set Method for Paraxial Geometrical Optics"],"prefix":"10.1137","volume":"28","author":[{"given":"Jianliang","family":"Qian","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shingyu","family":"Leung","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1995.1098"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0312(199911)52:11<1443::AID-CPA3>3.0.CO;2-Y"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025339522111"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2000.6530"},{"key":"R5","unstructured":"M. Born and E. Wolf,\n                      Principles of Optics\n                      , Macmillan, New York, 1964."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139993269242"},{"key":"R7","first-page":"757","volume":"134","author":"Burridge R.","year":"1998","journal-title":"Geophys. J. Internat.","ISSN":"https:\/\/id.crossref.org\/issn\/0956-540X","issn-type":"print"},{"key":"R8","unstructured":"V. Cerveny, I. A. Molotkov, and I. Psencik,\n                      Ray Method in Seismology\n                      , Univerzita Karlova Press, Praha, Czech Republic, 1977."},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1997.5721"},{"key":"R10","unstructured":"L.\u2010T. Cheng, H. Liu, and S. J. Osher,\n                      High Frequency Wave Propagation in Schrodinger Equations Using the Level Set Method\n                      , preprint 2003;"},{"key":"R10","unstructured":"available online at www.levelset.com."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1983-0690039-8"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(96)00023-4"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492902000119"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7033"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-003-0975-5"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6236"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.102476599"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1190\/1.1443439"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7085"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.12.004"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1190\/1.1443639"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1137\/S106482759732455X"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(03)00124-6"},{"key":"R24","unstructured":"S. Jin and S. Osher,\n                      A Level Set Method for the Computation of Multivalued Solutions to Quasi\u2010linear Hyperbolic PDEs and Hamilton\u2010Jacobi Equations\n                      , preprint 2003;"},{"key":"R24","unstructured":"available online at www.levelset.com."},{"key":"R25","unstructured":"Joseph Keller, Robert Lewis, Asymptotic methods for partial differential equations: the reduced wave equation and Maxwell\u2019s equations, Surveys Appl. Math., Vol. 1, Plenum, New York, 1995, 1\u20138297d:35020"},{"key":"R26","volume-title":"Generalized solutions of Hamilton\u2010Jacobi equations","author":"Lions Pierre\u2010Louis","year":"1982"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160190207"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1190\/1.1444814"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.2002.7080"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1007\/b98879"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/0728049"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6345"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1016\/S0165-2125(02)00101-4"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.12.017"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1190\/1.1451472"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6223"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1999.6389"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1994.1155"},{"key":"R39","unstructured":"W. W. Symes,\n                      Mathematics of Reflection Seismology\n                      , Annual Report, The Rice Inversion Project, Rice University, Houston, TX, 1995;"},{"key":"R39","unstructured":"also available online from http:\/\/www.trip.caam.rice.edu\/."},{"key":"R40","unstructured":"W. W. Symes,\n                      A slowness matching finite difference method for traveltimes beyond transmission caustics\n                      , Expanded Abstracts, in the 68th Annual International Meeting, Society of Exploration Geophysicists, 1998, pp. 1945\u20131948."},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1023\/A:1025380731197"},{"key":"R42","unstructured":"W. W. Symes, R. Versteeg, A. Sei, and Q. H. Tran,\n                      Kirchhoff Simulation, Migration and Inversion Using Finite Difference Traveltimes and Amplitudes\n                      , Annual Report, The Rice Inversion Project, Rice University, Houston, TX, 1994;"},{"key":"R42","unstructured":"also available online from http:\/\/www.trip.caam.rice.edu\/."},{"key":"R43","unstructured":"R. Thom,\n                      Structural Stability and Morphogenesis\n                      , Benjamin, New York, 1975."},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1190\/1.1443099"},{"key":"R45","first-page":"2062","volume":"78","author":"Vidale J.","year":"1988","journal-title":"Bull. Seismol. Soc. Amer.","ISSN":"https:\/\/id.crossref.org\/issn\/0037-1106","issn-type":"print"},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1190\/1.1443499"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1137\/0144010"},{"key":"R48","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112099004565"},{"key":"R49","unstructured":"L. Zhang,\n                      Imaging by the wavefront propagation method\n                      , Ph.D. thesis, Stanford University, Stanford, CA 94305, 1993."},{"key":"R50","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.0167"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/030601673","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:44:29Z","timestamp":1787334269000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/030601673"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,1]]},"references-count":54,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2006,1]]}},"alternative-id":["10.1137\/030601673"],"URL":"https:\/\/doi.org\/10.1137\/030601673","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2006,1]]}}}