{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,9]],"date-time":"2026-01-09T03:26:10Z","timestamp":1767929170233,"version":"3.49.0"},"reference-count":25,"publisher":"Society of Exploration Geophysicists","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2003,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Nonhyperbolic (long-spread) moveout provides essential information for a number of seismic inversion\/ processing applications, particularly for parameter estimation in anisotropic media. Here, we present an analytic expression for the quartic moveout coefficient A4 that controls the magnitude of nonhyperbolic moveout of pure (nonconverted) modes. Our result takes into account reflection-point dispersal on irregular interfaces and is valid for arbitrarily anisotropic, heterogeneous media. All quantities needed to compute A4 can be evaluated during the tracing of the zero-offset ray, so long-spread moveout can be modeled without time-consuming multioffset, multiazimuth ray tracing.<\/jats:p>\n                  <jats:p>The general equation for the quartic coefficient is then used to study azimuthally varying nonhyperbolic moveout of P-waves in a dipping transversely isotropic (TI) layer with an arbitrary tilt \u03bd of the symmetry axis. Assuming that the symmetry axis is confined to the dip plane, we employed the weak-anisotropy approximation to analyze the dependence of A4 on the anisotropic parameters. The linearized expression for A4 is proportional to the anellipticity coefficient \u03b7 \u2248 \u220a \u2212 \u03b4 and does not depend on the individual values of the Thomsen parameters. Typically, the magnitude of nonhyperbolic moveout in tilted TI media above a dipping reflector is highest near the reflector strike, whereas deviations from hyperbolic moveout on the dip line are substantial only for mild dips.<\/jats:p>\n                  <jats:p>The azimuthal variation of the quartic coefficient is governed by the tilt \u03bd and reflector dip \u03d5 and has a much more complicated character than the NMO\u2013velocity ellipse. For example, if the symmetry axis is vertical (VTI media, \u03bd = 0) and the dip \u03d5 &amp;gt;, 30\u00b0, A4 goes to zero on two lines with different azimuths where it changes sign. If the symmetry axis is orthogonal to the reflector (this model is typical for thrust-and-fold belts), the strike-line quartic coefficient is defined by the well-known expression for a horizontal VTI layer (i.e., it is independent of dip), while the dip-line A4 is proportional to cos 4\u03d5 and rapidly decreases with dip. The high sensitivity of the quartic moveout coefficient to the parameter \u03b7 and the tilt of the symmetry axis can be exploited in the inversion of wide-azimuth, long-spread P-wave data for the parameters of TI media.<\/jats:p>","DOI":"10.1190\/1.1620634","type":"journal-article","created":{"date-parts":[[2003,9,25]],"date-time":"2003-09-25T18:44:27Z","timestamp":1064515467000},"page":"1600-1610","source":"Crossref","is-referenced-by-count":52,"title":["Quartic moveout coefficient: 3D description and application to tilted TI media"],"prefix":"10.1190","volume":"68","author":[{"given":"Andres","family":"Pech","sequence":"first","affiliation":[{"name":"\u2217 Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Green Center, Golden, Colorado 80401. E-mail: ilya@dix.mines.edu."}]},{"given":"Ilya","family":"Tsvankin","sequence":"additional","affiliation":[{"name":"\u2217 Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Green Center, Golden, Colorado 80401. E-mail: ilya@dix.mines.edu."}]},{"given":"Vladimir","family":"Grechka","sequence":"additional","affiliation":[{"name":"\u2021 Formerly Colorado School of Mines, Center for Wave Phenomena, Department of Geophysics, Golden, Colorado 80401; presently Shell International Exploration & Production Inc., Bellaire Technology Center, 3737 Bellaire Blvd., Houston, Texas 77001-0481. E-mail: vladimir.grechka@shell.com."}]}],"member":"186","published-online":{"date-parts":[[2003,1,1]]},"reference":[{"key":"2025121112501960400_R1","doi-asserted-by":"crossref","first-page":"1738","DOI":"10.1190\/1.1444469","article-title":"Nonhyperbolic reflection moveout for horizontal transverse isotropy","volume":"63","author":"Al-Dajani","year":"1998","journal-title":"Geophysics"},{"key":"2025121112501960400_R2","doi-asserted-by":"crossref","first-page":"1839","DOI":"10.1190\/1.1444285","article-title":"Velocity analysis using nonhyperbolic moveout in transversely isotropic media","volume":"62","author":"Alkhalifah","year":"1997","journal-title":"Geophysics"},{"key":"2025121112501960400_R3","doi-asserted-by":"crossref","first-page":"1550","DOI":"10.1190\/1.1443888","article-title":"Velocity analysis for transversely isotropic media","volume":"60","author":"Alkhalifah","year":"1995","journal-title":"Geophysics"},{"key":"2025121112501960400_R4","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1190\/1.1442826","article-title":"Seismic traveltime inversion for transverse isotropy","volume":"55","author":"Byun","year":"1990","journal-title":"Geophysics"},{"issue":"12","key":"2025121112501960400_R5","first-page":"91","article-title":"Interpretation of effective common-depth-point parameters for a spatial system of homogeneous beds with curved boundaries","volume":"20","author":"Chernjak","year":"1979","journal-title":"Soviet Geology and Geophysics"},{"key":"2025121112501960400_R6","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1190\/1.1444321","article-title":"A convenient expression for the NMO velocity function in terms of ray parameter","volume":"63","author":"Cohen","year":"1998","journal-title":"Geophysics"},{"key":"2025121112501960400_R7","author":"Fomel","year":"2001"},{"key":"2025121112501960400_R8","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1190\/1.1444407","article-title":"Feasibility of nonhyperbolic moveout inversion in transversely isotropic media","volume":"63","author":"Grechka","year":"1998","journal-title":"Geophysics"},{"key":"2025121112501960400_R9","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1190\/1.1444386","article-title":"3-D description of normal moveout in anisotropic inhomogeneous media","volume":"63","author":"Grechka","year":"1998","journal-title":"Geophysics"},{"key":"2025121112501960400_R10","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1190\/1.1444714","article-title":"Inversion of azimuthally dependent NMO velocity in transversely isotropic media with a tilted axis of symmetry","volume":"65","author":"Grechka","year":"2000","journal-title":"Geophysics"},{"key":"2025121112501960400_R11","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1190\/1.1484536","article-title":"NMO-velocity surfaces and Dix-type formulas in heterogeneous anisotropic media","volume":"67","author":"Grechka","year":"2002","journal-title":"Geophysics"},{"key":"2025121112501960400_R12","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1046\/j.1365-2478.1999.00120.x","article-title":"Generalized Dix equation and analytic treatment of normal-moveout velocity for anisotropic media","volume":"47","author":"Grechka","year":"1999","journal-title":"Geophys. 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