×

zbMATH — the first resource for mathematics

An adjoint-based Jacobi-type iterative method for elastic full waveform inversion problem. (English) Zbl 1410.86015
Summary: Full waveform inversion (FWI) is a promising technique that is capable of creating high-resolution subsurface images of the earth from seismic data. However, it is computationally expensive, especially when 3D elastic wave equation is considered. In this paper an adjoint-based computational algorithm has been proposed to address the computational challenges. The important strategy in this work is to decouple the two Lamé parameters so that two half-sized subproblems are resulted and solved separately. Mathematically, the inverse problem is formulated as a PDE-constrained optimization problem in which the objective functional is defined as the misfit between observational and synthetic data. The parameters to be recovered are the spatially varying Lamé parameters which are of great interests to geophysicists for the purpose of hydrocarbon exploration and subsurface imaging. The gradient of the misfit functional with respect to the Lamé parameters is calculated by solving the adjoint elastic wave equation, whilst the Quasi-Newton method (L-BFGS) is used to minimize the misfit functional. Numerical experiments demonstrated that the new method is accurate, efficient and robust in recovering Lamé parameters from seismic data.
MSC:
86A15 Seismology (including tsunami modeling), earthquakes
86-08 Computational methods for problems pertaining to geophysics
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Abdoulaev, Gassan S.; Ren, Kui; Hielscher, Andreas H., Optical tomography as a PDE-constrained optimization problem, Inverse Prob., 21, 5, 1507, (2005) · Zbl 1086.35116
[2] Bischof, Christian, ADIFOR 2.0: automatic differentiation of Fortran 77 programs, Comput. Sci. Eng., IEEE, 3.3, 3, 18-32, (1996)
[3] Biros, George; Ghattas, Omar, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. part I: the Krylov-Schur solver, SIAM J. Sci. Comput., 27, 2, 687-713, (2005) · Zbl 1091.65061
[4] George, Biros; Ghattas, Omar, Parallel Lagrange-Newton-Krylov-Schur methods for PDE-constrained optimization. part II: the Lagrange-Newton solver and its application to optimal control of steady viscous flows, SIAM J. Sci. Comput., 27, 2, 714-739, (2005) · Zbl 1091.65062
[5] Brossier, Romain; Operto, Stéphane; Virieux, Jean, Which data residual norm for robust elastic frequency-domain full waveform inversion?, Geophysics, 75, 3, R37-R46, (2010)
[6] Danping, Cao; Liao, Wenyuan, An adjoint-based hybrid computational method for crosswell seismic inversion, Comput. Sci. Eng., (2014), <http://doi.ieeecomputersociety.org/10.1109/MCSE.2014> · Zbl 1344.86002
[7] Danping, Cao; Liao, Wenyuan, A computational method for full waveform inversion of crosswell seismic data using automatic differentiation, Comput. Phys. Commun., (2014) · Zbl 1344.86002
[8] Castellanos Clara, Algorithmic and methodological developments towards full waveform inversion in 3D elastic media, Expanded Abstracts of 81st Annual Internat SEG Mtg, 2011, 2793-2798.
[9] Charpentier, Isabelle; Ghemires, Mohammed, Efficient adjoint derivatives: application to the meteorological model meso-NH, Opt. Meth. Softw., 13, 1, 35-63, (2000) · Zbl 0983.76067
[10] Charpentier, I., Checkpointing schemes for adjoint codes: application to the meteorological model meso-NH, SIAM J. Sci. Comput., 22, 6, 2135-2151, (2001) · Zbl 0989.86003
[11] Fichtner, Andreas, Full seismic waveform modelling and inversion, (2011), Springer Berlin/Heidelberg
[12] Giering, Ralf; Kaminski, Thomas, Recipes for adjoint code construction, ACM Trans. Math. Soft. (TOMS), 24, 4, 437-474, (1998) · Zbl 0934.65027
[13] Giering, Ralf; Kaminski, Thomas, Applying TAF to generate efficient derivative code of Fortran 7795 programs, PAMM, 2, 1, 54-57, (2003)
[14] Mark Gockenbach, S., Efficient and automatic implementation of the adjoint state method, ACM Trans. Math. Softw. (TOMS), 28, 1, 22-44, (2002) · Zbl 1070.65560
[15] Laurent, Hascoet; Pascual, Valérie, The tapenade automatic differentiation tool: principles, model and specification, ACM Trans. Math. Softw. (TOMS), 39, 3, 20, (2013) · Zbl 1295.65026
[16] Patrick, Lailly, The seismic inverse problem as a sequence of before stack migrations, Conference on Inverse Scattering: Theory and Application, (1983), Society for Industrial and Applied Mathematics Philadelphia, PA
[17] Leung, Shingyu; Qian, Jianliang, An adjoint state method for three-dimensional transmission traveltime tomography using first-arrivals, Commun. Math. Sci., 4, 1, 249-266, (2006) · Zbl 1096.65062
[18] Liao, Wenyuan, A computational method to estimate the unknown coefficient in a wave equation using boundary measurements, Inverse Prob. Sci. Eng., 19, 6, 855-877, (2011) · Zbl 1252.65160
[19] Liu, Dong C.; Nocedal, Jorge, On the limited memory BFGS method for large scale optimization, Math. Programming, 45, 1-3, 503-528, (1989) · Zbl 0696.90048
[20] Liu, Qinya; Tromp, Jeroen, Finite-frequency kernels based on adjoint methods, Bull. Seismol. Soc. Am., 96, 6, 2383-2397, (2006)
[21] Nocedal, Jorge, Updating quasi-Newton matrices with limited storage, Math. Comput., 35, 151, 773-782, (1980) · Zbl 0464.65037
[22] Bjrn, Sjögreen; Anders Petersson, N., A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation, J. Sci. Comput., 52, 1, 17-48, (2012) · Zbl 1255.65162
[23] Plessix, R.-E., A review of the adjoint-state method for computing the gradient of a functional with geophysical applications, Geophys. J. Int., 167, 2, 495-503, (2006)
[24] Gerhard, Pratt R.; Shin, Changsoo; Hick, G. J., Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion, Geophys. J. Int., 133, 2, 341-362, (1998)
[25] Gerhard, Pratt R., Seismic waveform inversion in the frequency domain, part 1: theory and verification in a physical scale model, Geophysics, 64, 3, 888-901, (1999)
[26] Rees, Tyrone; Sue Dollar, H.; Wathen, Andrew J., Optimal solvers for PDE-constrained optimization, SIAM J. Sci. Comput., 32, 1, 271-298, (2010) · Zbl 1208.49035
[27] Anouar, Romdhane, Shallow-structure characterization by 2D elastic full-waveform inversion, Geophysics, 76, 3, R81-R93, (2011)
[28] Rudin, Leonid I.; Osher, Stanley; Fatemi, Emad, Nonlinear total variation based noise removal algorithms, Phys. D, 60, 1, 259-268, (1992) · Zbl 0780.49028
[29] Yousef Saad. Iterative methods for sparse linear systems. SIAM, 2003. · Zbl 1031.65046
[30] Sen Mrinal, K.; Stoffa, Paul L., Global optimization methods in geophysical inversion, (1995), Elsevier · Zbl 0871.90107
[31] Taillandier, Cédric, First-arrival traveltime tomography based on the adjoint-state method, Geophysics, 74, 6, WCB1-WCB10, (2009)
[32] Tarantola, Albert, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 8, 1259-1266, (1984)
[33] Tarantola, Albert, A strategy for nonlinear elastic inversion of seismic reflection data, Geophysics, 51, 10, 1893-1903, (1986)
[34] Tran, Khiem T.; Hiltunen, Dennis R., Two-dimensional inversion of full waveforms using simulated annealing, J. Geotech. Geoenviron. Eng., 138, 9, 1075-1090, (2011)
[35] Virieux, J.; Operto, S., An overview of full-waveform inversion in exploration geophysics, Geophys. Today, 74, 6, 383, (2010)
[36] M. Warner, J.V. Morgan, Three-dimensional anisotropic acoustic and elastic full-waveform seismic inversion, AGU Fall Meeting Abstracts, vol. 1, 2013.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.