Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
International Journal for Uncertainty Quantification
IF: 4.911 5-Year IF: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2019027745
pages 395-414


Mohammad Motamed
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico
Daniel Appelo
Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado


We present a Bayesian framework based on a new exponential likelihood function driven by the quadratic Wasserstein metric. Compared to conventional Bayesian models based on Gaussian likelihood functions driven by the least-squares norm (L2 norm), the new framework features several advantages. First, the new framework does not rely on the like-lihood of the measurement noise and hence can treat complicated noise structures such as combined additive and multiplicative noise. Second, unlike the normal likelihood function, the Wasserstein-based exponential likelihood function does not usually generate multiple local extrema. As a result, the new framework features better convergence to correct posteriors when a Markov Chain Monte Carlo sampling algorithm is employed. Third, in the particular case of signal processing problems, although a normal likelihood function measures only the amplitude differences between the observed and simulated signals, the new likelihood function can capture both amplitude and phase differences. We apply the new framework to a class of signal processing problems, that is, the inverse uncertainty quantification of waveforms, and demonstrate its advantages compared to Bayesian models with normal likelihood functions.


  1. Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B., Bayesian Data Analysis, New York: Chapman and Hall/CRC, 2004.

  2. Kaipo, J. and Somersalo, E., Statistical and Computational Inverse Problems, New York: Springer, 2005.

  3. Stuart, A.M., Inverse Problems: A Bayesian Perspective, Acta Numer, 19:451-559,2010.

  4. Gamerman, D. and Lopes, H.F., Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, New York: Chapman and Hall/CRC, 2006.

  5. Villani, C., Topics in Optimal Transportation, Vol. 58, Graduate Studies in Mathematics, Providence, RI: American Mathematical Society, 2003.

  6. Villani, C., Optimal Transport: Old and New, Vol. 338, Comprehensive Studies in Mathematics, Berlin: Springer-Verlag, 2009.

  7. Engquist,B. andFroese, B.D., Application of the Wasserstein Metric to Seismic Signals, Commun. Math. Sci., 12(5):979-988, 2014.

  8. Engquist, B., Brittany, B.F., and Yang, Y., Optimal Transport for Seismic Full Waveform Inversion, Commun. Math. Sci, 14(8):2309-2330,2016.

  9. Yang, Y., Engquist, B., Sun, J., and Hamfeldt, B.F., Application of Optimal Transport and the Quadratic Wasserstein Metric to Full-Waveform Inversion, Geophys., 83(1):R43-R62,2018.

  10. Engquist, B. and Yang, Y., Seismic Imaging and Optimal Transport, Commun. Inf. Syst., arXiv:1808.04801,2018.

  11. Chen, J., Chen, Y., Wu, H., and Yang, D., The Quadratic Wasserstein Metric for Earthquake Location, J. Comput. Phys, 373:188-209,2018.

  12. Ballesio, M., Beck, J., Pandey, A., Parisi, L., von Schwerin, E., and Tempone, R., Multilevel Monte Carlo Acceleration of Seismic Wave Propagation under Uncertainty, Numer Anal., arXiv:1810.01710,2018.

  13. Metivier, L., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., Measuring the Misfit between Seismograms Using an Optimal Transport Distance: Application to Full Waveform Inversion, Geophys. J. Int., 205(1):345-377,2016.

  14. Metivier, L., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., An Optimal Transport Approach for Seismic Tomography: Application to 3D Full Waveform Inversion, Inverse Probl, 32(11): 115008,2016.

  15. Metivier, L., Allain, A., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., A Graph-Space Approach to Optimal Transport for Full Waveform Inversion, Soc. Explor. Geophys., pp. 1158-1162,2018.

  16. Metivier, L., Allain, A., Brossier, R., Merigot, Q., Oudet, E., and Virieux, J., Optimal Transport for Mitigating Cycle Skipping in Full-Waveform Inversion: A Graph-Space Transform Approach, Geophys., 83(5):R515-R540,2018.

  17. Hedjazian, N., Bodin, T., and Metivier, L., An Optimal Transport Approach to Linearized Inversion of Receiver Functions, Geophys. J. Int., 216(1): 130-147,2019.

  18. Bayes, T., Price, R., and Canton, J., An Essay towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, F.R.S. Communicated by Mr. Price, in a Letter to John Canton, A.M.F.R., Philos. Trans. R S. London, 53:370-418, 1763.

  19. Goodman, J., Speckle Phenomena in Optics: Theory and Applications, Englewood, CO: Roberts and Company Publ., 2007.

  20. Papadakis, N., Optimal Transport for Image Processing, PhD, Universite de Bordeaux, 2015.

  21. Kolouri, S., Zou, Y., and Rohde, G.K., Sliced Wasserstein Kernels for Probability Distributions, Proc. of 2016IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5258-5267,2016.

  22. Esfahani, P.M. and Kuhn, D., Data-Driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations, Math. Prog;, 171:115-166,2017.

  23. Benamou, J.D., Froese, B.D., and Oberman, A.M., Numerical Solution of the Optimal Transportation Problem Using the Monge-Ampere Equation, J. Comput. Phys, 260:107-126,2014.

  24. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., and Teller, E., Equation of State Calculations by Fast Computing Machines, J. Chem. Phys, 21:1087-1092,1953.

  25. Hastings, W.K., Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika, 57:97-109,1970.

  26. Chib, S. and Greenberg, E., Understanding the Metropolis-Hastings Algorithm, Am. Stat., 49:327-335,1995.

  27. Clayton, R. and Engquist, B., Absorbing Boundary Conditions for Acoustic and Elastic Wave Equations, Bull. Seismol. Soc. Am., 67:1529-1540,1977.

  28. Nilsson, S., Petersson, N.A., Sjogren, B., and Kreiss, H.O., Stable Difference Approximations for the Elastic Wave Equation in Second Order Formulation, SIAM J. Numer. Anal, 45:1902-1936,2007.

  29. Sjogreen, B. and Petersson, N.A., Source Estimation by Full Wave Form Inversion, J. Sci. Comput:., 59:247-276,2014.

  30. Long, Q., Motamed, M., and Tempone, R., Fast Bayesian Optimal Experimental Design for Seismic Source Inversion, Comput. Methods Appl. Mech. Eng., 291:123-145,2015.

  31. Motamed, M., Nobile, F., and Tempone, R., A Stochastic Collocation Method for the Second Order Wave Equation with a Discontinuous Random Speed, Numer. Math., 123:493-536,2013.

  32. Appelo, D., Hagstrom, T., and Vargas, A., Hermite Methods for the Scalar Wave Equation, SIAM J. Sci. Comput, 40(6):A3902-A3927,2018.

  33. Blei, D.M., Kucukelbir, A., and McAuliffe, J.D., Variational Inference: A Review for Statisticians, J. Am. Stat. Assoc., 112:859-877,2017.

Articles with similar content:

Hybrid Methods in Engineering, Vol.2, 2000, issue 1
M. de la Sen
Bayesian Network as Instrument of Intelligent Data Analysis
Journal of Automation and Information Sciences, Vol.39, 2007, issue 8
Alexander N. Terentyev, Petr I. Bidyuk, L. A. Korshevnyuk
International Journal for Uncertainty Quantification, Vol.5, 2015, issue 2
Yong Huang, James L. Beck
International Journal for Uncertainty Quantification, Vol.3, 2013, issue 4
Alexandros Taflanidis, James L. Beck
International Journal for Uncertainty Quantification, Vol.1, 2011, issue 2
Sankaran Mahadevan, Bin Liang