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Uncertainty quantification for inverse problems with weak partial-differential-equation constraints

Authors:

In statistical inverse problems, the objective is a complete statistical description of unknown parameters from noisy observations to quantify uncertainties in unknown parameters. We consider inverse problems with partial-differential-equation (PDE) constraints, which are applicable to many seismic problems. Bayesian inference is one of the most widely used approaches to precisely quantify statistics through a posterior distribution, incorporating uncertainties in observed data, modeling kernel, and prior knowledge of parameters. Typically when formulating the posterior distribution, the PDE constraints are required to be exactly satisfied, resulting in a highly nonlinear forward map and a posterior distribution with many local maxima. These drawbacks make it difficult to find an appropriate approximation for the posterior distribution. Another complicating factor is that traditional Markov chain Monte Carlo (MCMC) methods are known to converge slowly for realistically sized problems. To overcome these drawbacks, we relax the PDE constraints by introducing an auxiliary variable, which allows for Gaussian errors in the PDE and yields a bilinear posterior distribution with weak PDE constraints that is more amenable to uncertainty quantification because of its special structure. We determine that for a particular range of variance choices for the PDE misfit term, the new posterior distribution has fewer modes and can be well-approximated by a Gaussian distribution, which can then be sampled in a straightforward manner. Because it is prohibitively expensive to explicitly construct the dense covariance matrix of the Gaussian approximation for problems with more than 105 unknowns, we have developed a method to implicitly construct it, which enables efficient sampling. We apply this framework to 2D seismic inverse problems with 1800 and 92,455 unknown parameters. The results illustrate that our framework can produce comparable statistical quantities with those produced by conventional MCMC-type methods while requiring far fewer PDE solves, which are the main computational bottlenecks in these problems.

REFERENCES

  • Amestoy, P., R. Brossier, A. Buttari, J.-Y. L’Excellent, T. Mary, L. Métivier, A. Miniussi, and S. Operto, 2016, Fast 3D frequency-domain full-waveform inversion with a parallel block low-rank multifrontal direct solver: Application to OBC data from the North Sea: Geophysics, 81, no. 6, R363–R383, doi: 10.1190/geo2016-0052.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Amestoy, P. R., I. S. Duff, J.-Y. L’Excellent, and J. Koster, 2001, A fully asynchronous multifrontal solver using distributed dynamic scheduling: SIAM Journal on Matrix Analysis and Applications, 23, 15–41, doi: 10.1137/S0895479899358194.SJMAEL0895-4798CrossrefWeb of ScienceGoogle Scholar
  • Amestoy, P. R., A. Guermouche, J.-Y. L’Excellent, and S. Pralet, 2006, Hybrid scheduling for the parallel solution of linear systems: Parallel Computing, 32, 136–156, doi: 10.1016/j.parco.2005.07.004.PACOEJ0167-8191CrossrefWeb of ScienceGoogle Scholar
  • Bardsley, J. M., A. Seppänen, A. Solonen, H. Haario, and J. Kaipio, 2015, Randomize-then-optimize for sampling and uncertainty quantification in electrical impedance tomography: SIAM/ASA Journal on Uncertainty Quantification, 3, 1136–1158, doi: 10.1137/140978272.CrossrefGoogle Scholar
  • Bardsley, J. M., A. Solonen, H. Haario, and M. Laine, 2014, Randomize-then-optimize: A method for sampling from posterior distributions in nonlinear inverse problems: SIAM Journal on Scientific Computing, 36, A1895–A1910, doi: 10.1137/140964023.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Bayes, T., R. Price, and J. Canton, 1763, An essay towards solving a problem in the doctrine of chances: Philosophical Transactions, 53, 370–418, doi: 10.1098/rstl.1763.0053.CrossrefGoogle Scholar
  • Biegler, L. T., T. F. Coleman, A. Conn, and F. N. Santosa, 2012, Large-scale optimization with applications. Part I: Optimization in inverse problems and design: Springer Science and Business Media.Google Scholar
  • Bui-Thanh, T., O. Ghattas, J. Martin, and G. Stadler, 2013, A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion: SIAM Journal on Scientific Computing, 35, A2494–A2523, doi: 10.1137/12089586X.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Bunks, C., F. M. Saleck, S. Zaleski, and G. Chavent, 1995, Multiscale seismic waveform inversion: Geophysics, 60, 1457–1473, doi: 10.1190/1.1443880.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Chen, Y., and D. S. Oliver, 2012, Ensemble randomized maximum likelihood method as an iterative ensemble smoother: Mathematical Geosciences, 44, 1–26, doi: 10.1007/s11004-011-9376-z.CrossrefWeb of ScienceGoogle Scholar
  • Chen, Z., D. Cheng, W. Feng, and T. Wu, 2013, An optimal 9-point finite difference scheme for the Helmholtz equation with PML: International Journal of Numerical Analysis and Modeling, 10, 389–410.Web of ScienceGoogle Scholar
  • Dummit, D. S., and R. M. Foote, 2004, Abstract algebra: Wiley Hoboken.Google Scholar
  • Efron, B., 1981, Nonparametric estimates of standard error: The jackknife, the bootstrap and other methods: Biometrika, 68, 589–599, doi: 10.1093/biomet/68.3.589.BIOKAX0006-3444CrossrefWeb of ScienceGoogle Scholar
  • Efron, B., 1992, Bootstrap methods: Another look at the jackknife, in S. KotzN. L. Johnson, eds., Breakthroughs in statistics: Springer, 569–593.CrossrefGoogle Scholar
  • Ely, G., A. Malcolm, and O. V. Poliannikov, 2017, Assessing uncertainties in velocity models and images with a fast nonlinear uncertainty quantification method: Geophysics, 83, no. 2, R63–R75, doi: 10.1190/geo2017-0321.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Engquist, B., and B. D. Froese, 2014, Application of the Wasserstein metric to seismic signals: Communications in Mathematical Sciences, 12, 979–988, doi: 10.4310/CMS.2014.v12.n5.a7.1539-6746CrossrefWeb of ScienceGoogle Scholar
  • Epanomeritakis, I., V. Akçelik, O. Ghattas, and J. Bielak, 2008, A Newton-CG method for large-scale three-dimensional elastic full-waveform seismic inversion: Inverse Problems, 24, 034015, doi: 10.1088/0266-5611/24/3/034015.INPEEY0266-5611CrossrefWeb of ScienceGoogle Scholar
  • Esser, E., L. Guasch, T. van Leeuwen, A. Y. Aravkin, and F. J. Herrmann, 2018, Total-variation regularization strategies in full-waveform inversion: SIAM Journal on Imaging Sciences, 11, 376–406, doi: 10.1137/17M111328X.CrossrefWeb of ScienceGoogle Scholar
  • Evensen, G., 2009, Data assimilation: The ensemble Kalman filter: Springer Science and Business Media.CrossrefGoogle Scholar
  • Fang, Z., C. Lee, C. Da Silva, F. J. Herrmann, and R. Kuske, 2015, Uncertainty quantification for wavefield reconstruction inversion: 77th Annual International Conference and Exhibition, EAGE, Extended Abstracts, doi: 10.3997/2214-4609.201413198.CrossrefGoogle Scholar
  • Fang, Z., C. Y. Lee, C. Da Silva, F. J. Herrmann, and T. Van Leeuwen, 2016, Uncertainty quantification for wavefield reconstruction inversion using a PDE-free semidefinite Hessian and randomize-then-optimize method: 86th Annual International Meeting, SEG, Expanded Abstracts, 1390–1394, doi: 10.1190/segam2016-13879108.1.AbstractGoogle Scholar
  • Fichtner, A., B. L. Kennett, H. Igel, and H.-P. Bunge, 2008, Theoretical background for continental- and global-scale full-waveform inversion in the time-frequency domain: Geophysical Journal International, 175, 665–685, doi: 10.1111/j.1365-246x.2008.03923.x.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Fichtner, A., and J. Trampert, 2011, Resolution analysis in full waveform inversion: Geophysical Journal International, 187, 1604–1624, doi: 10.1111/j.1365-246x.2011.05218.x.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Fichtner, A., and T. van Leeuwen, 2015, Resolution analysis by random probing: Journal of Geophysical Research: Solid Earth, 120, 5549–5573.CrossrefWeb of ScienceGoogle Scholar
  • Fisher, M., M. Leutbecher, and G. Kelly, 2005, On the equivalence between Kalman smoothing and weak-constraint four-dimensional variational data assimilation: Quarterly Journal of the Royal Meteorological Society, 131, 3235–3246, doi: 10.1256/qj.04.142.QJRMAM0035-9009CrossrefWeb of ScienceGoogle Scholar
  • Gee, L. S., and T. H. Jordan, 1992, Generalized seismological data functionals: Geophysical Journal International, 111, 363–390, doi: 10.1111/j.1365-246x.1992.tb00584.x.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Golub, G., and V. Pereyra, 2003, Separable nonlinear least squares: The variable projection method and its applications: Inverse Problems, 19, R1, doi: 10.1088/0266-5611/19/2/201.INPEEY0266-5611CrossrefWeb of ScienceGoogle Scholar
  • Golub, G. H., and C. F. Van Loan, 2012, Matrix computations: Johns Hopkins University Press.Google Scholar
  • Gouveia, W. P., and J. A. Scales, 1998, Bayesian seismic waveform inversion: Parameter estimation and uncertainty analysis: Journal of Geophysical Research: Solid Earth, 103, 2759–2779, doi: 10.1029/97JB02933.CrossrefWeb of ScienceGoogle Scholar
  • Haario, H., M. Laine, A. Mira, and E. Saksman, 2006, Dram: Efficient adaptive MCMC: Statistics and Computing, 16, 339–354, doi: 10.1007/s11222-006-9438-0.STACE30960-3174CrossrefWeb of ScienceGoogle Scholar
  • Haber, E., U. M. Ascher, and D. Oldenburg, 2000, On optimization techniques for solving nonlinear inverse problems: Inverse Problems, 16, 1263–1280, doi: 10.1088/0266-5611/16/5/309.INPEEY0266-5611CrossrefWeb of ScienceGoogle Scholar
  • Hastings, W. K., 1970, Monte Carlo sampling methods using Markov chains and their applications: Biometrika, 57, 97–109, doi: 10.1093/biomet/57.1.97.BIOKAX0006-3444CrossrefWeb of ScienceGoogle Scholar
  • Hinze, M., R. Pinnau, M. Ulbrich, and S. Ulbrich, 2008, Optimization with PDE constraints: Springer Science and Business Media.Google Scholar
  • Huang, G., R. Nammour, and W. Symes, 2017, Full-waveform inversion via source-receiver extension: Geophysics, 82, no. 3, R153–R171, doi: 10.1190/geo2016-0301.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Kaipio, J., and E. Somersalo, 2006, Statistical and computational inverse problems: Springer Science and Business Media.Google Scholar
  • Keilis-Borok, V. I., and T. B. Yanovskaja, 1967, Inverse problems of seismology (structural review): Geophysical Journal International of the Royal Astronomical Society, 13, 223–234, doi: 10.1111/j.1365-246X.1967.tb02156.x.GJOUDQ0275-9128CrossrefGoogle Scholar
  • Kitanidis, P. K., 1995, Recent advances in geostatistical inference on hydrogeological variables: Reviews of Geophysics, 33, 1103–1109, doi: 10.1029/95RG00183.REGEEP8755-1209CrossrefWeb of ScienceGoogle Scholar
  • Li, X., A. Y. Aravkin, T. van Leeuwen, and F. J. Herrmann, 2012, Fast randomized full-waveform inversion with compressive sensing: Geophysics, 77, no. 3, A13–A17, doi: 10.1190/geo2011-0410.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Li, Y., B. Biondi, R. Clapp, and D. Nichols, 2014, Wave-equation migration velocity analysis for VTI models: Geophysics, 79, no. 3, WA59–WA68, doi: 10.1190/geo2013-0338.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Lieberman, C., K. Willcox, and O. Ghattas, 2010, Parameter and state model reduction for large-scale statistical inverse problems: SIAM Journal on Scientific Computing, 32, 2523–2542, doi: 10.1137/090775622.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Lim, S., 2017, Bayesian inverse problems and seismic inversion: Ph.D. thesis, University of Oxford.Google Scholar
  • Luo, Y., and G. T. Schuster, 1991, Wave-equation traveltime inversion: Geophysics, 56, 645–653, doi: 10.1190/1.1443081.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Martin, J., L. C. Wilcox, C. Burstedde, and O. Ghattas, 2012, A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion: SIAM Journal on Scientific Computing, 34, A1460–A1487, doi: 10.1137/110845598.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Matheron, G., 2012, Estimating and choosing: An essay on probability in practice: Springer Science and Business Media.Google Scholar
  • Métivier, L., R. Brossier, Q. Mérigot, E. Oudet, and J. Virieux, 2016, Measuring the misfit between seismograms using an optimal transport distance: Application to full waveform inversion: Geophysical Journal International, 205, 345–377, doi: 10.1093/gji/ggw014.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Métivier, L., R. Brossier, J. Virieux, and S. Operto, 2013, Full waveform inversion and the truncated Newton method: SIAM Journal on Scientific Computing, 35, B401–B437, doi: 10.1137/120877854.SJOCE31064-8275CrossrefWeb of ScienceGoogle Scholar
  • Metropolis, N., A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, 1953, Equation of state calculations by fast computing machines: The Journal of Chemical Physics, 21, 1087–1092, doi: 10.1063/1.1699114.CrossrefWeb of ScienceGoogle Scholar
  • Mosegaard, K., and A. Tarantola, 1995, Monte Carlo sampling of solutions to inverse problems: Journal of Geophysical Research: Solid Earth, 100, 12431–12447, doi: 10.1029/94JB03097.CrossrefWeb of ScienceGoogle Scholar
  • Nocedal, J., and S. J. Wright, 2006, Numerical optimization: Springer-Verlag.CrossrefGoogle Scholar
  • Orieux, F., O. Féron, and J.-F. Giovannelli, 2012, Sampling high-dimensional Gaussian distributions for general linear inverse problems: IEEE Signal Processing Letters, 19, 251–254, doi: 10.1109/LSP.2012.2189104.CrossrefWeb of ScienceGoogle Scholar
  • Osypov, K., Y. Yang, A. Fournier, N. Ivanova, R. Bachrach, C. E. Yarman, Y. You, D. Nichols, and M. Woodward, 2013, Model-uncertainty quantification in seismic tomography: Method and applications: Geophysical Prospecting, 61, 1114–1134, doi: 10.1111/1365-2478.12058.GPPRAR0016-8025CrossrefWeb of ScienceGoogle Scholar
  • Papandreou, G., and A. L. Yuille, 2010, Gaussian sampling by local perturbations: Advances in Neural Information Processing Systems, 1858–1866.1049-5258Google Scholar
  • 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, doi: 10.1111/j.1365-246x.2006.02978.x.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Pratt, G., C. Shin, and G. Hicks, 1998, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion: Geophysical Journal International, 133, 341–362, doi: 10.1046/j.1365-246X.1998.00498.x.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Pratt, R. G., 1999, Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model: Geophysics, 64, 888–901, doi: 10.1190/1.1444597.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Press, F., 1968, Earth models obtained by Monte Carlo inversion: Journal of Geophysical Research, 73, 5223–5234, doi: 10.1029/JB073i016p05223.JGREA20148-0227CrossrefWeb of ScienceGoogle Scholar
  • Roberts, G. O., and J. S. Rosenthal, 2001, Optimal scaling for various Metropolis-Hastings algorithms: Statistical Science, 16, 351–367, doi: 10.1214/ss/1015346320.STSCEP0883-4237CrossrefWeb of ScienceGoogle Scholar
  • Rue, H., 2001, Fast sampling of Gaussian Markov random fields: Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63, 325–338, doi: 10.1111/1467-9868.00288.CrossrefWeb of ScienceGoogle Scholar
  • Sava, P., and B. Biondi, 2004, Wave-equation migration velocity analysis. I: Theory: Geophysical Prospecting, 52, 593–606, doi: 10.1111/j.1365-2478.2004.00447.x.GPPRAR0016-8025CrossrefWeb of ScienceGoogle Scholar
  • Sirgue, L., and R. G. Pratt, 2004, Efficient waveform inversion and imaging: A strategy for selecting temporal frequencies: Geophysics, 69, 231–248, doi: 10.1190/1.1649391.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Solonen, A., A. Bibov, J. M. Bardsley, and H. Haario, 2014, Optimization-based sampling in ensemble Kalman filtering: International Journal for Uncertainty Quantification, 4, 349–364, doi: 10.1615/Int.J.UncertaintyQuantification.v4.i4.CrossrefWeb of ScienceGoogle Scholar
  • Stuart, G., W. Yang, S. Minkoff, and F. Pereira, 2016, A two-stage Markov chain Monte Carlo method for velocity estimation and uncertainty quantification: 86th Annual International Meeting, SEG, Expanded Abstracts, 3682–3687, doi: 10.1190/segam2016-13865449.1.AbstractGoogle Scholar
  • Tarantola, A., 2005, Inverse problem theory and methods for model parameter estimation: SIAM.CrossrefGoogle Scholar
  • Tarantola, A., and B. Valette, 1982a, Generalized nonlinear inverse problems solved using the least squares criterion: Reviews of Geophysics, 20, 219–232, doi: 10.1029/RG020i002p00219.REGEEP8755-1209CrossrefWeb of ScienceGoogle Scholar
  • Tarantola, A., and B. Valette, 1982b, Inverse problems = quest for information: Journal of Geophysics, 50, 159–170.JGEOD40340-062XWeb of ScienceGoogle Scholar
  • Thurin, J., R. Brossier, and L. Métivier, 2017, An ensemble-transform Kalman filter: Full-waveform inversion scheme for uncertainty estimation: 87th Annual International Meeting, SEG, Expanded Abstracts, 1307–1313, doi: 10.1190/segam2017-17733053.1.AbstractGoogle Scholar
  • van Leeuwen, T., A. Y. Aravkin, and F. J. Herrmann, 2014, Comment on: “Application of the variable projection scheme for frequency-domain full-waveform inversion” (M. Li, J. Rickett, and A. Abubakar, Geophysics, 78, no. 6, R249-R257): Geophysics, 79, no. 3, X11–X17, doi: 10.1190/geo2013-0466.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • van Leeuwen, T., and F. J. Herrmann, 2013a, Fast waveform inversion without source encoding: Geophysical Prospecting, 61, 10–19, doi: 10.1111/j.1365-2478.2012.01096.x.GPPRAR0016-8025CrossrefWeb of ScienceGoogle Scholar
  • van Leeuwen, T., and F. J. Herrmann, 2013b, Mitigating local minima in full-waveform inversion by expanding the search space: Geophysical Journal International, 195, 661–667, doi: 10.1093/gji/ggt258.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • van Leeuwen, T., and F. J. Herrmann, 2015, A penalty method for PDE-constrained optimization in inverse problems: Inverse Problems, 32, 015007, doi: 10.1088/0266-5611/32/1/015007.INPEEY0266-5611CrossrefWeb of ScienceGoogle Scholar
  • van Leeuwen, T., and W. Mulder, 2010, A correlation-based misfit criterion for wave-equation traveltime tomography: Geophysical Journal International, 182, 1383–1394, doi: 10.1111/j.1365-246X.2010.04681.x.GJINEA0956-540XCrossrefWeb of ScienceGoogle Scholar
  • Virieux, J., and S. Operto, 2009, An overview of full-waveform inversion in exploration geophysics: Geophysics, 74, no. 6, WCC1–WCC26, doi: 10.1190/1.3238367.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Warner, M., and L. Guasch, 2016, Adaptive waveform inversion: Theory: Geophysics, 81, no. 6, R429–R445, doi: 10.1190/geo2015-0387.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Warner, M., T. Nangoo, N. Shah, A. Umpleby, and J. Morgan, 2013, Full-waveform inversion of cycle-skipped seismic data by frequency down-shifting: 83rd Annual International Meeting, SEG, Expanded Abstracts, 903–907, doi: 10.1190/segam2013-1067.1.AbstractGoogle Scholar
  • Yang, Y., B. Engquist, J. Sun, and B. F. Hamfeldt, 2018, Application of optimal transport and the quadratic Wasserstein metric to full-waveform inversion: Geophysics, 83, no. 1, R43–R62, doi: 10.1190/geo2016-0663.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar
  • Yuan, S., and S. Wang, 2013, Spectral sparse Bayesian learning reflectivity inversion: Geophysical Prospecting, 61, 735–746, doi: 10.1111/1365-2478.12000.GPPRAR0016-8025CrossrefWeb of ScienceGoogle Scholar
  • Yuan, S., S. Wang, M. Ma, Y. Ji, and L. Deng, 2017, Sparse Bayesian learning-based time-variant deconvolution: IEEE Transactions on Geoscience and Remote Sensing, 55, 6182–6194, doi: 10.1109/TGRS.2017.2722223.IGRSD20196-2892CrossrefWeb of ScienceGoogle Scholar
  • Zhu, H., S. Li, S. Fomel, G. Stadler, and O. Ghattas, 2016, A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration: Geophysics, 81, no. 5, R307–R323, doi: 10.1190/geo2015-0641.1.GPYSA70016-8033AbstractWeb of ScienceGoogle Scholar