Abstract

We propose an extended full-waveform inversion formulation that includes general convex constraints on the model. Though the full problem is highly nonconvex, the overarching optimization scheme arrives at geologically plausible results by solving a sequence of relaxed and warm-started constrained convex subproblems. The combination of box, total variation, and successively relaxed asymmetric total variation constraints allows us to steer free from parasitic local minima while keeping the estimated physical parameters laterally continuous and in a physically realistic range. For accurate starting models, numerical experiments carried out on the challenging 2004 BP velocity benchmark demonstrate that bound and total variation constraints improve the inversion result significantly by removing inversion artifacts, related to source encoding, and by clearly improved delineation of top, bottom, and flanks of a high-velocity high-contrast salt inclusion. The experiments also show that for poor starting models these two constraints by themselves are insufficient to detect the bottom of high-velocity inclusions such as salt. Inclusion of the one-sided asymmetric total variation constraint overcomes this issue by discouraging velocity lows to buildup during the early stages of the inversion. To the best of the authors' knowledge the presented algorithm is the first to successfully remove the imprint of local minima caused by poor starting models and band-width limited finite aperture data.

Keywords

  1. full-waveform inversion
  2. total variation
  3. regularization
  4. constraints
  5. optimization

MSC codes

  1. 35
  2. 86

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
A. Tarantola and A. Valette, Generalized nonlinear inverse problems solved using the least squares criterion, Rev. Geophys. Space Phys., 20 (1982), pp. 129--232.
2.
A. Tarantola, Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49 (1984), pp. 1259--1266, https://doi.org/10.1190/1.1441754.
3.
W. W. Symes, The seismic reflection inverse problem, Inverse Problems, 25 (2009), 123008, https://doi.org/10.1088/0266-5611/25/12/123008.
4.
W. W. Symes, Migration velocity analysis and waveform inversion, Geophys. Prospect., 56 (2008), pp. 765--790, https://doi.org/10.1111/j.1365-2478.2008.00698.x.
5.
B. Biondi and A. Almomin, Tomographic full-waveform inversion (TFWI) by combining FWI and wave-equation migration velocity analysis, The Leading Edge, 32 (2013), pp. 1074--1080, https://doi.org/10.1190/tle32091074.1.
6.
M. Warner and L. Guasch, Adaptive waveform inversion-fwi without cycle skipping-theory, in Proceedings of the 76th EAGE Conference and Exhibition, 2014.
7.
T. van Leeuwen and F. J. Herrmann, Mitigating local minima in full-waveform inversion by expanding the search space, Geophys. J. Int., 195 (2013), pp. 661--667, https://doi.org/10.1093/gji/ggt258.
8.
T. van Leeuwen and F. J. Herrmann, A penalty method for PDE-constrained optimization in inverse problems, Inverse Problems, 32 (2016), 015007, https://doi.org/10.1088/0266-5611/32/1/015007.
9.
A. Y. Aravkin and T. van Leeuwen, Estimating nuisance parameters in inverse problems, Inverse Problems, 28 (2012), 115016.
10.
A. Y. Aravkin, D. Drusvyatskiy, and T. van Leeuwen, Efficient quadratic penalization through the partial minimization technique, IEEE Trans. Automat. Control, 99 (2017), p. 1.
11.
D. P. Bertsekas, Nonlinear Programming, 2nd ed., Athena Scientific, Belmont, MA, 1999.
12.
S. Bonettini, R. Zanella, and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002.
13.
M. Schmidt, D. Kim, and S. Sra, Projected Newton-type methods in machine learning, in Optimization for Machine Learning, MIT Press, Cambrigde, MA, 2011, pp. 305--330.
14.
J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer Ser. Oper. Res. Financ. Eng., P. Glynn and S. M. Robinson, eds., Springer, New York, 2006.
15.
E. Esser, Y. Lou, and J. Xin, A method for finding structured sparse solutions to nonnegative least squares problems with applications, SIAM J. Imaging Sci., 6 (2013), pp. 2010--2046, https://doi.org/10.1137/13090540X.
16.
L. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), pp. 259--268.
17.
E. T. Chung, T. F. Chan, and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization, J. Comput. Phys., 205 (2005), pp. 357--372.
18.
V. Akcelik, G. Biros, and O. Ghattas, Parallel multiscale Gauss-Newton-Krylov methods for inverse wave propagation, in Proceedings of the 2002 ACM/IEEE Conference on Supercomputing, 2002, IEEE, Los Alamitos, CA, pp. 1--15.
19.
A. Guitton, Blocky regularization schemes for full-waveform inversion, Geophys. Prospect., 60 (2012), pp. 870--884, https://doi.org/10.1111/j.1365-2478.2012.01025.x.
20.
Z. Guo and M. de Hoop, Shape optimization in full waveform inversion with sparse blocky model representations, Proceedings of the Project Review, 1 (2012), pp. 189--208, http://gmig.math.purdue.edu/pdfs/2012/12-12.pdf.
21.
M. Maharramov and B. Biondi, Robust Joint Full-Waveform Inversion of Time-Lapse Seismic Data Sets with Total-Variation Regularization, preprint, https://arxiv.org/abs/1408.0645, 2014.
22.
A. Bourgeois, M. Bourget, P. Lailly, M. Poulet, P. Ricarte, and R. Versteeg, The Marmousi experience, in Proceedings of the 1990 EAEG Workshop on Practical Aspects of Seismic Data Inversion, Eur. Assoc. Expl. Geophys., 1991, pp. 5--16.
23.
M. Zhu and T. Chan, An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration, UCLA CAM Report [08-34], UCLA, Los Angeles, CA, 2008, pp. 1--29.
24.
E. Esser, X. Zhang, and T. F. Chan, A general framework for a class of first order primal-dual algorithms for convex optimization in imaging science, SIAM J. Imaging Sci., 3 (2010), pp. 1015--1046, https://doi.org/10.1137/09076934X.
25.
A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), pp. 120--145.
26.
B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective, SIAM J. Imaging Sci., 5 (2012), pp. 119--149, https://doi.org/10.1137/100814494.
27.
X. Zhang, M. Burger, and S. Osher, A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46 (2010), pp. 20--46.
28.
P. Brucker, An O(n) algorithm for quadratic knapsack problems, Oper. Res. Lett., 3 (1984), pp. 163--166.
29.
T. Goldstein, E. Esser, and R. Baraniuk, Adaptive Primal-Dual Hybrid Gradient Methods for Saddle-Point Problems, preprint, https://arxiv.org/abs/1305.0546, 2013.
30.
C. Bunks, F. M. Saleck, S. Zaleski, and G. Chavent, Multiscale seismic waveform inversion, Geophysics, 60 (1995), pp. 1457--1473, https://doi.org/10.1190/1.1443880.
31.
F. Billette and S. Brandsberg-Dahl, The 2004 BP velocity benchmark, in Proceedings of the 67th EAGE Conference and Exhibition, 2005, pp. 13--16.
32.
J. R. Krebs, J. E. Anderson, D. Hinkley, R. Neelamani, S. Lee, A. Baumstein, and M.-D. Lacasse, Fast full wave seismic inversion using source encoding, Geophysics, 74 (2009), pp. WCC177--WCC188, https://doi.org/10.1190/1.3230502.
33.
E. Haber, M. Chung, and F. J. Herrmann, An effective method for parameter estimation with PDE constraints with multiple right-hand sides, SIAM J. Optim., 22 (2012), pp. 739--757, https://doi.org/10.1137/11081126X.
34.
M. P. Friedlander and M. Schmidt, Hybrid deterministic-stochastic methods for data fitting, SIAM J. Sci. Comput., 34 (2012), pp. A1380--A1405, https://doi.org/10.1137/110830629.
35.
T. van Leeuwen and F. J. Herrmann, Fast waveform inversion without source-encoding, Geophys. Prospect., 61 (2013), pp. 10--19, http://doi.org/10.1111/j.1365-2478.2012.01096.x.
36.
B. Peters, F. J. Herrmann, and T. van Leeuwen, Wave-equation based inversion with the penalty method: Adjoint-state versus wavefield-reconstruction inversion, in Proceedings of the 76th EAGE Conference and Exhibition, 2014, https://doi.org/10.3997/2214-4609.20140704.
37.
E. Esser, L. Guasch, F. J. Herrmann, and M. Warner, Constrained waveform inversion for automatic salt flooding, The Leading Edge, 35 (2016), pp. 235--239, https://doi.org/10.1190/tle35030235.1.
38.
E. Haber, U. M. Ascher, and D. Oldenburg, On optimization techniques for solving nonlinear inverse problems, Inverse Problems, 16 (2000), pp. 1263--1280.
39.
J. Virieux and S. Operto, An overview of full-waveform inversion in exploration geophysics, Geophysics, 74 (2009), pp. WCC1--WCC26, https://doi.org/10.1190/1.3238367.
40.
M. Fisher, M. Leutbecher, and G. A. Kelly, On the equivalence between Kalman smoothing and weak-constraint four-dimensional variational data assimilation, Q. J. R. Meteorol. Soc., 131 (2005), pp. 3235--3246, https://doi.org/10.1256/qj.04.142.

Information & Authors

Information

Published In

cover image SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
Pages: 376 - 406
ISSN (online): 1936-4954

History

Submitted: 24 January 2017
Accepted: 18 October 2017
Published online: 7 February 2018

Keywords

  1. full-waveform inversion
  2. total variation
  3. regularization
  4. constraints
  5. optimization

MSC codes

  1. 35
  2. 86

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media