none 10.1190/1.1487129 Society of Exploration Geophysicists Society of Exploration Geophysicists 186 4728782 5981 2025121112413609700 10.1190 2025-12-11T17:41:45Z 2002-10-11T15:23:40Z 147 Geophysics 1942-2156 0016-8033 2001 01 01 2001 66 6 Changsoo Shin ∗Seoul National University, School of Civil, Urban and Geosystem Engineering, San 56–1, Sinlim-dong, Kwanak-ku, Seoul, 151–740, Korea. E-mail: css@model.snu.ac.kr. Kwangjin Yoon ‡Shin's Imaging Technology, Inc., 910 Owner Venture Town, 1661–4, Bongcheon 7-dong, Kwanak-ku, Seoul 151–057, Korea. E-mail: kjyoon@gpl.snu.ac.kr; dwyang@gpl.snu.ac.kr; hrlim@hanmail.net. Kurt J. Marfurt ∗∗University of Houston, Allied Geophysical Laboratories, 4800 Calhoun Rd., Houston, Texas 77205. E-mail: kmarfurt@uh.edu. Keunpil Park §Korea Institute of Geology, Mining & Materials, Petroleum & Marine Resource Division, 30 Kajung-dong, Yusung-ku, Taejon 305–350, Korea. E-mail: kpp@kigmm.re.kr; bigchung@rock25t.kigam.re.kr. Dongwoo Yang ‡Shin's Imaging Technology, Inc., 910 Owner Venture Town, 1661–4, Bongcheon 7-dong, Kwanak-ku, Seoul 151–057, Korea. E-mail: kjyoon@gpl.snu.ac.kr; dwyang@gpl.snu.ac.kr; hrlim@hanmail.net. Harry Y. Lim ‡Shin's Imaging Technology, Inc., 910 Owner Venture Town, 1661–4, Bongcheon 7-dong, Kwanak-ku, Seoul 151–057, Korea. E-mail: kjyoon@gpl.snu.ac.kr; dwyang@gpl.snu.ac.kr; hrlim@hanmail.net. Seunghwan Chung §Korea Institute of Geology, Mining & Materials, Petroleum & Marine Resource Division, 30 Kajung-dong, Yusung-ku, Taejon 305–350, Korea. E-mail: kpp@kigmm.re.kr; bigchung@rock25t.kigam.re.kr. Sungryul Shin §§Korea Maritime University, Department of Energy and Resources Engineering, 1 Dongsam-dong, Youngdo-ku, Busan 606–791, Korea. E-mail: srshin@hanara.kmaritime.ac.kr. Abstract Linearized inversion of surface seismic data for a model of the earth's subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth's subsurface. This sensitivity, or Jacobian, matrix is usually quite expensive to estimate for all but the simplest model parameterizations. We exploit the numerical structure of the finite-element method, modern sparse matrix technology, and source–receiver reciprocity to develop an algorithm that explicitly calculates the Jacobian matrix at only the cost of a forward model solution. Furthermore, we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the Hessian matrix, as predicted by the classical Gauss-Newton method. Our method is applicable to the full suite of wave scattering problems amenable to finite-element forward modeling. We demonstrate our method through some simple 2-D synthetic examples. 2001 01 01 2001 06 08 2001 1856 1863 10.1190/1.1487129 https://pubs.geoscienceworld.org/geophysics/article/66/6/1856/107192/Efficient-calculation-of-a-partial-derivative https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/66/6/1856/3693354/gsgpy_66_6_1856.pdf https://pubs.geoscienceworld.org/seg/geophysics/article-pdf/66/6/1856/3693354/gsgpy_66_6_1856.pdf http://geophysics.geoscienceworld.org/cgi/doi/10.1190/1.1487129 Geophysics Baysal 48 1514 1983 10.1190/1.1441434 Reverse time migration J. Acoust. Soc. Am. Berkhout 93 2017 1993 10.1121/1.406715 A unified approach to acoustical reflection imaging, part II: The inverse problem Geophysics Bleistein 52 931 1987 10.1190/1.1442363 On the imaging of reflectors in the earth The Marmousi experience, Proceedings of the 1990 EAEG workshop on Practical Aspects of Seismic Data Inversion Bourgeois 5 1991 Marmousi, model and data Geophysics Chavent 60 476 1995 10.1190/1.1443785 Determination of background velocities by multiple migration fitting Geophysics Ehinger 61 1813 1996 10.1190/1.1444097 Green's function implementation of common-offset, wave-equation migration Geophysics Gazdag 49 124 1984 10.1190/1.1441643 Migration of seismic data by phase-shift plus interpolation Geophys. J. Internat. Geller 115 699 1993 10.1111/j.1365-246X.1993.tb01488.x Two efficient algorithms for iterative linearized inversion of seismic waveform data Geophysics Gray 62 929 1997 10.1190/1.1444200 True-amplitude seismic migration: A comparison of three approaches Geophys. Prosp. Hagedoorn 2 85 1954 10.1111/j.1365-2478.1954.tb01281.x A process of seismic reflection interpretation Geophys. J. Internat. Jin 108 575 1992 10.1111/j.1365-246X.1992.tb04637.x Two dimensional asymptotic iterative elastic inversion The seismic inverse problem as a sequence of before stack migration Lailly 206 1984 Geophysics Lambaré 57 1138 1992 10.1190/1.1443328 Iterative asymptotic inversion in the acoustic approximation Supercomputers in seismic exploration Marfurt 203 1989 The future of iterative modeling of geophysical exploration Geophys. J. Roy. Astr. Soc. McMechan 71 613 1983 10.1111/j.1365-246X.1982.tb02788.x Migration by exploration of time dependent boundary values Geophysics Mora 52 1211 1987 10.1190/1.1442384 Nonlinear two-dimensional elastic inversion of multi-offset seismic data Geophys. J. Internat. Pratt 133 341 1998 10.1046/j.1365-246X.1998.00498.x Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion Nonlinear elastic wave inversion by blocky parameterization Shin 1988 Geophysics Shin 64 1774 1999 10.1190/1.1444683 Understanding CMP stacking hyperbola in terms of partial derivative wavefield Geophysics Stork 57 680 1992 10.1190/1.1443282 Reflection tomography in the postmigrated domain Geophysics Tarantola 49 1259 1984 10.1190/1.1441754 Inversion of seismic reflection data in the acoustic approximation Iterative depth migration by backward time propagation Whitmore 827 1983 J. Geophys. Res. Woodhouse 89 5953 1984 10.1029/JB089iB07p05953 Mapping the upper mantle: Three-dimensional modeling of earth structure by inversion of seismic waveforms The finite element method Zienkiewicz 3rd ed. 1977