Abstract
We analyze different misfit functions for comparing synthetic and observed data in seismic imaging, for example, the Wasserstein metric and the conventional least-squares norm. We revisit the convexity and insensitivity to noise of the Wasserstein metric which demonstrate the robustness of the metric in seismic inversion. Numerical results illustrate that full waveform inversion with quadratic Wasserstein metric can often effectively overcome the risk of local minimum trapping in the optimization part of the algorithm. A mathematical study on Fréchet derivative with respect to the model parameters of the objective functions further illustrates the role of optimal transport maps in this iterative approach. In this context we refer to the objective function as misfit. A realistic numerical example is presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Alford, R., Kelly, K., Boore, D.M.: Accuracy of finite-difference modeling of the acoustic wave equation. Geophysics 39(6), 834–842 (1974)
Billette, F., Brandsberg-Dahl, S.: The 2004 BP velocity benchmark. In: 67th EAGE Conference & Exhibition (2005)
Engquist, B., Froese, B.D.: Application of the Wasserstein metric to seismic signals. Commun. Math. Sci. 12(5), 979–988 (2014)
Engquist, B., Froese, B.D., Yang, Y.: Optimal transport for seismic full waveform inversion. Commun. Math. Sci. 14(8), 2309–2330 (2016)
Huang, G., Wang, H., Ren, H.: Two new gradient precondition schemes for full waveform inversion. arXiv preprint arXiv:1406.1864 (2014)
Liu, J., Chauris, H., Calandra, H.: The normalized integration method-an alternative to full waveform inversion? In: 25th Symposium on the Application of Geophpysics to Engineering & Environmental Problems (2012)
Métivier, L., Brossier, R., Mrigot, Q., Oudet, E., Virieux, J.: An optimal transport approach for seismic tomography: application to 3D full waveform inversion. Inverse Prob. 32(11), 115008 (2016)
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)
Stolt, R.H., Weglein, A.B.: Seismic Imaging and Inversion: Volume 1: Application of Linear Inverse Theory, vol. 1. Cambridge University Press, Cambridge (2012)
Tarantola, A., Valette, B.: Generalized nonlinear inverse problems solved using the least squares criterion. Rev. Geophys. 20(2), 219–232 (1982)
Villani, C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence (2003)
Virieux, J., Operto, S.: An overview of full-waveform inversion in exploration geophysics. Geophysics 74(6), WCC1–WCC26 (2009)
Yang, Y., Engquist, B., Sun, J., Froese, B.D.: Application of optimal transport and the quadratic wasserstein metric to full-waveform inversion. arXiv preprint arXiv:1612.05075 (2016)
Acknowledgments
We thank Sergey Fomel, Junzhe Sun and Zhiguang Xue for helpful discussions, and thank the sponsors of the Texas Consortium for Computational Seismology (TCCS) for financial support. This work was also partially supported by NSF DMS-1620396.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Yang, Y., Engquist, B. (2017). Analysis of Optimal Transport Related Misfit Functions in Seismic Imaging. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2017. Lecture Notes in Computer Science(), vol 10589. Springer, Cham. https://doi.org/10.1007/978-3-319-68445-1_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-68445-1_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68444-4
Online ISBN: 978-3-319-68445-1
eBook Packages: Computer ScienceComputer Science (R0)