Abstract In full waveform inversion (FWI), Hessian information of the misfit function is of vital importance for accelerating the convergence of the inversion; however, it usually is not feasible to directly calculate the Hessian matrix and its inverse. Although the limited memory Broyden–Fletcher–Goldfarb–Shanno (L–BFGS) or Hessian-free inexact Newton (HFN) methods are able to use approximate Hessian information, the information they collect is limited. The two methods can be interlaced because they are able to provide Hessian information for each other; however, the performance of the hybrid iterative method is dependent on the effective switch between the two methods. We have designed a new scheme to realize the dynamic switch between the two methods based on the decrease ratio (DR) of the misfit function (objective function), and we propose a modified hybrid iterative optimization method. In the new scheme, we compare the DR of the two methods for a given computational cost, and choose the method with a faster DR. Using these steps, the modified method always implements the most efficient method. The results of Marmousi and overthrust model testings indicate that the convergence with our modified method is significantly faster than that in the L-BFGS method with no loss of inversion quality. Moreover, our modified outperforms the enriched method by a little speedup of the convergence. It also exhibits better efficiency than the HFN method.
Full waveform inversion|Hessian information|limited memory BFGS method|Hessian-free inexact Newton method|decrease ratio
About author: Wang Yi received his MS degree (2009) from the School of Mathematics and Statistics at Wuhan University. He is now a PhD student at School of Ocean and Earth Science at Tongji University, majoring in solid geophysics. His main focus is the research of full waveform inversion.
Cite this article:
WANG Yi,DONG Liang-Guo,LIU Yu-Zhu. Improved hybrid iterative optimization method for seismic full waveform inversion[J]. APPLIED GEOPHYSICS, 2013, 10(3): 265-277.
[1]
Akcelik, V., 2002, Multiscale Newton-Krylov methods for inverse acoustic wave propagation: PhD thesis. Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA.
[2]
Bae, H. S., Pyun, S., Chung, W., Kang, S. G., and Shin, C., 2012, Frequency-domain acoustic-elastic coupled waveform inversion using the Gauss-Newton conjugate gradient method: Geophysical Prospecting, 60, 413 - 432.
[3]
Berenger, J. P., 1994, A perfectly matched layer for the absorption of electromagnetic waves: J. Comput. Phys, 114, 185 - 200.
[4]
Berkhout, A. J., 2012, Combining full wavefield migration and full waveform inversion, a glance into the future of seismic imaging: Geophysics, 77, S43 - S50.
[5]
Boonyasiriwat, C., Schuster, G. T., Paul, V., and Cao, W. P., 2010, Applications of multiscale waveform inversion to marine data using a flooding technique and dynamic early-arrival windows: Geophysics, 75, R129 - R136.
[6]
Brossier, R., Operto, S., and Virieux, J., 2009, Seismic imaging of complex structures by 2D elastic frequency domain full-waveform inversion: Geophysics,74, WCC105 - WCC118.
[7]
Bunks, C. F., Salek, M., Zaleski, S., and Chavent, G., 1995, Multiscale seismic waveform inversion: Geophysics, 60, 1457 - 1473.
[8]
Choi, Y., Min, D. J., and Shin, C., 2008, Two-dimensional waveform inversion of multi-component data in acoustic-elastic coupled media: Geophysical Prospecting, 56, 863 - 881.
[9]
Crase, E., Pica, A., Noble, M., McDonald, J., and Tarantola, A., 1990, Robust elastic non-linear waveform inversion: Application to real data: Geophysics, 55, 527 - 538.
[10]
Epanomeritakis, I., Akcelik, V., Ghattas, O., and Bielak, J., 2008, A Newton-CG method for large-scale three-dimensional elastic full waveform seismic inversion: Inverse Problems, 24, 1 - 26.
[11]
Fichtner, A., 2011, Full Seismic Waveform Modelling and Inversion: Springer, Berlin, German, 141 - 146.
[12]
Fichtner, A., and Trampert, J., 2011, Hessian kernels of seismic data functionals based upon adjoint techniques: Geophysical Journal International, 185, 775 - 798.
[13]
Guitton, A., and Symes, W. W., 2003, Robust inversion of seismic data using the Huber norm: Geophysics, 68, 1310 - 1319.
[14]
Hu, W., Abubakar, A., Habashy, T. M., and Liu, J., 2011, Preconditioned non-linear conjugate gradient method for frequency domain full-waveform seismic inversion: Geophysical Prospecting, 59, 477 - 491.
[15]
Kelley, C. T., 1999, Iterative methods for optimization: SIAM, Philadelphia, USA, 71 - 83.
[16]
Lailly, P., 1983, The seismic inverse problem as a sequence of before stack migrations: Conference on Inverse Scattering: Theory and Application, (Eds. Bednar, J.B. and Weglein, A.) Society for Industrial and Applied Mathematics, 206 - 220.
[17]
Liu, D. C., and Nocedal, J., 1989, On the limited memory BFGS method for large scale optimization: - Mathematical Programming, 45, 503 - 528.
[18]
Metivier, L., Brossier, R., Virieux, J., and Operto, S., 2012a, Toward Gauss-Newton and exact Newton optimization for full waveform inversion: 74th EAGE Conference and Exhibition, Copenhagen, Denmark, Expanded Abstracts, P016.
[19]
Metivier, L., Brossier, R., Virieux, J., and Operto, S., 2012b, The truncated Newton method for Full Waveform Inversion: 82nd SEG Annual Meeting, Las Vegas, USA, Expanded Abstracts, 0981.1
[20]
Mora, P. R., 1987, Nonlinear two-dimensional elastic inversion of multi-offset seismic data: Geophysics, 52, 1211 - 1228.
[21]
Morales, J. L., and Nocedal, J., 2000, Automatic preconditioning by limited memory quasi-Newton updates: SIAM Journal on Optimization, 10, 1079 - 1096.
[22]
Morales, J. L., and Nocedal, J., 2002, Enriched methods for large-scale unconstrained optimization: Computational Optimization and Applications, 21, 143 - 154.
[23]
Nash, S.G, and Nocedal, J., 1991, A numerical study of the limited memory BFGS method and the truncated-Newton method for large-scale optimization: SIAM Journal on Optimization, 1, 358 - 372.
[24]
Nocedal, J., 1980, Updating quasi-Newton matrices with limited storage: Mathematics of Computation, 35, 773 - 782.
[25]
Nocedal, J., and Wright, S. J., 2006, Numerical optimization (2nd edition): Springer, New York, USA, 101 - 184 .
[26]
Pica, A., Diet, J., and Tarantola, A., 1990, Nonlinear inversion of seismic reflection data in laterally invariant medium. Geophysics, 55, 284 - 292.
[27]
Plessix, R. E., 2006, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications: Geophysical Journal International, 167, 495 - 503.
[28]
Pratt, R. G., Shin, C., and Hicks, G. J., 1998, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion: Geophysical Journal International, 133, 341 - 362.
[29]
Shan, R., Bian, A. F., Yu, W. H., and Qu, Z., 2011, Pre-stack full waveform inversion for reservoir prediction: Oil Geophysical Prospecting ( in Chinese ), 46, 629 - 633.
[30]
Sheen, D. H., Tuncay, K., Baag, C. E., and Ortoleva, P. J., 2006, Time domain Gauss-Newton seismic waveform inversion in elastic media: Geophysical Journal International, 167, 1373 - 1384.
[31]
Shin, C., Jang, S., and Min, D. J., 2001, Improved amplitude preservation for prestack depth migration by inverse scattering theory: Geophysical Prospecting, 49, 592 - 606.
[32]
Tape, C., 2009, Seismic tomography of southern California using adjoint methods: PhD thesis. California Institute of Technology, Pasadena, California, USA.
[33]
Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation: Geophysics, 49, 1259 - 1266.
[34]
Tromp, J., Tape, C., and Liu, Q., 2005, Seismic tomography, adjoint methods, time reversal, and banana-doughnut kernels: Geophysical Journal International, 160, 195 - 216.
[35]
Virieux, J., and Operto, S., 2009, An overview of full-waveform inversion in exploration geophysics: Geophysics, 74, WCC1 - WCC26.
[36]
Yuan, Y. X., 2008, Computation Methods for Nonlinear Optimization: Science Press, Beijing, China, 82 - 88.
2013, Vol. 10 
