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THE WASSERSTEIN-FISHER-RAO METRIC FOR WAVEFORM BASED EARTHQUAKE LOCATION

Datong Zhou1, Jing Chen1, Hao Wu1, Dinghui Yang1, Lingyun Qiu2,3   

  1. 1. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;
    2. Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China;
    3. Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China
  • Received:2020-02-11 Revised:2021-05-17 Published:2023-03-14
  • Contact: Hao Wu, Email:hwu@tsinghua.edu.cn
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871297, 11971258, U1839206), the National Key Research and Development Program of China on Monitoring, Early Warning and Prevention of Major Natural Disaster (Grant No. 2017YFC1500301), and Tsinghua University Initiative Scientific Research Program. The authors are indebted to Prof. Y. Brenier for his helpful discussions.

Datong Zhou, Jing Chen, Hao Wu, Dinghui Yang, Lingyun Qiu. THE WASSERSTEIN-FISHER-RAO METRIC FOR WAVEFORM BASED EARTHQUAKE LOCATION[J]. Journal of Computational Mathematics, 2023, 41(3): 437-458.

In this paper, we apply the Wasserstein-Fisher-Rao (WFR) metric from the unbalanced optimal transport theory to the earthquake location problem. Compared with the quadratic Wasserstein (W2) metric from the classical optimal transport theory, the advantage of this method is that it retains the important amplitude information as a new constraint, which avoids the problem of the degeneration of the optimization objective function near the real earthquake hypocenter and origin time. As a result, the deviation of the global minimum of the optimization objective function based on the WFR metric from the true solution can be much smaller than the results based on the W2 metric when there exists strong data noise. Thus, we develop an accurate earthquake location method under strong data noise. Many numerical experiments verify our conclusions.

CLC Number: 

[1] J.D. Benamou and Y. Brenier, A computational fluid mechanics solution to the MongeKantorovich mass transfer problem, Numer. Math., 84(2000), 375-393.
[2] J.D. Benamou, G. Carlier, M. Cuturi, L. Nenna and G. Peyr, Iterative Bregman projections for regularized transportation problems, SIAM J. Sci. Comput., 37:2(2015), A1111-A1138.
[3] J.D. Bray, R.B. Seed and H.B. Seed, Analysis of earthquake fault rupture propagation through cohesive soil, J. Geotech. Engrg., 120:3(1994), 562-580.
[4] J. Chen, Y. Chen, H. Wu and D. Yang, The quadratic Wasserstein metric for Earthquake Location, J. Comput. Phys., 373(2018), 188-209.
[5] J. Chen, H. Jing, P. Tong, H. Wu and D. Yang, The auxiliary function method for waveform based earthquake location, J. Comput. Phys., 413(2020), 109453.
[6] J. Chen, S.K. Kufner, X. Yuan, B. Heit, H. Wu, D. Yang, B. Schurr, and S. Kay, Lithospheric Delamination Beneath the Southern Puna Plateau Resolved by Local Earthquake Tomography, J. Geophys. Res.:Solid Earth, 125(2020), e2019JB019040.
[7] L. Chizat, G. Peyré, B. Schmitzer and F.X. Vialard, An Interpolating Distance Between Optimal Transport and Fisher-Rao Metrics, Found. Comput. Math., 18:1(2018), 1-44.
[8] L. Chizat, G. Peyré, B. Schmitzer and F.X. Vialard, Scaling Algorithms for unbalanced Optimal Transport Problems, Math. Comput., 87(2018), 2563-2609.
[9] F.D. Col, M. Papadopoulou, E. Koivisto, Ƚ. Sito, M. Savolainen and L.V. Socco, Application of surface-wave tomography to mineral exploration:a case study from Siilinjärvi, Finland, Geophys. Prospect., 68:1(2020), 254-269.
[10] M.A. Dablain, The application of high-order differencing to the scalar wave equation, Geophysics, 51:1(1986), 54-66.
[11] B. Engquist and B.D. Froese, Application of the Wasserstein metric to seismic signals, Commun. Math. Sci., 12:5(2014), 979-988.
[12] B. Engquist, B.D. Froese and Y. Yang, Optimal transport for seismic full waveform inversion, Commun. Math. Sci., 14:8(2016), 2309-2330.
[13] B. Engquist, K. Ren and Y. Yang, The quadratic Wasserstein metric for inverse data matching, Inverse Probl., 36:5(2020), 055001.
[14] B. Engquist and Y. Yang, Seismic imaging and optimal transport, Commun. Inf. Syst., 19:2(2019), 95-145.
[15] T.O. Gallouët, M. Laborde and L. Monsaingeon, An unbalanced Optimal Transport splitting scheme for general advection-reaction-diffusion problems, ESAIM:COCV, 25(2019), 8.
[16] W. Gangbo, W. Li, S. Osher and M. Puthawala, Unnormalized optimal transport, J. Comput. Phys., 399(2019), 108940.
[17] M.C. Ge, Analysis of source location algorithms Part I:Overview and non-iterative methods, J. Acoust. Emiss., 21(2003), 14-28.
[18] S. Kondratyev, L. Monsaingeon and D. Vorotnikov, A new optimal transport distance on the space of finite Radon measures, Adv. Differ. Equat., 21(2016), 1117-1164.
[19] D. Komatitsch and J. Tromp, A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation, Geophys. J. Int., 154(2003), 146-153.
[20] J.J. Kosowsky and A.L. Yuille, The invisible hand algorithm:Solving the assignment problem with statistical physics, Neural Netw., 7:3(1994), 477-490.
[21] T. Le, M. Yamada, K. Fukumizu and M. Cuturi, Tree-Sliced Variants of Wasserstein Distances, arXiv:1902.00342v3, 2019.
[22] J. Li, D. Yang, H. Wu and X. Ma, A low-dispersive method using the high-order stereo-modelling operator for solving 2-D wave equations, Geophys. J. Int., 210(2017), 1938-1964.
[23] R. Li and F. Yang, A reconstructed discontinuous approximation to Monge-Ampere equation in least square formation, arXiv:2010.09921v3, 2019.
[24] M. Liero, A. Mielke and G. Savaré, Optimal transport in competition with reaction:the HellingerKantorovich distance and geodesic curves, SIAM J. Math. Analysis, 48:4(2016), 2869-2911.
[25] M. Liero, A. Mielke and G. Savaré, Optimal Entropy-Transport problems and a new HellingerKantorovich distance between positive measures, Invent. Math., 211(2018) 969-1117.
[26] Q. Liu, J. Polet, D. Komatitsch and J. Tromp, Spectral-Element Moment Tensor Inversion for Earthquakes in Southern California, Bull. seism. Soc. Am., 94:5(2004), 1748-1761.
[27] R. Madariaga, Seismic Source Theory, in Treatise on Geophysics (Second Edition), S. Gerald (ed.), Elsevier B.V., 2015, 51-71.
[28] C. Meng, Y. Ke, J. Zhang, M. Zhang, W. Zhong and P. Ma, Large-scale optimal transport map estimation using projection pursuit, Adv. Neural Inf. Process. Syst., 32(2019), 8118-8129.
[29] C. Meng, J. Yu, J. Zhang, P. Ma and W. Zhong, Sufficient dimension reduction for classification using principal optimal transport direction, arXiv:2010.09921v3, 2020.
[30] L. Métivier, R. Brossier, Q. Mérigot, E. Oudet and J. Virieux, Measuring the misfit between seismograms using an optimal transport distance:application to full waveform inversion, Geophys. J. Int., 205(2016), 345-377.
[31] L. Métivier, R. Brossier, Q. Mérigot, E. Oudet and J. Virieux, An optimal transport approach for seismic tomography:application to 3D full waveform inversion, Inverse Probl., 32(2016), 115008.
[32] W. Pan and Y. Wang, On the influence of different misfit functions for attenuation estimation in viscoelastic full-waveform inversion:synthetic study, Geophys. J. Int., 221:2(2020), 1292-1319.
[33] G. Peyr and M. Cuturi, Computational Optimal Transport:With Applications to Data Science, Found. Trends Mach. Learn., 11:5-6(2019), 355-607.
[34] B. Piccoli and F. Rossi, Generalized Wasserstein Distance and its Application to Transport Equations with Source, Arch. Rational Mech. Anal., 211(2014), 335-358.
[35] L. Qiu, J. Ramos-Martnez, A. Valenciano, Y. Yang and B. Engquist, Full-waveform inversion with an exponentially encoded optimal-transport norm, SEG Tech. Program Expanded Abstr., (2017) 1286-1290.
[36] N. Rawlinson, S. Pozgay and S. Fishwick, Seismic tomography:A window into deep Earth, Phys. Earth Planet. Inter., 178(2010), 101-135.
[37] F. Santambrogio, Optimal Transport for Applied Mathematicians, Birkhäuser, 2015.
[38] C. Satriano, A. Lomax and A. Zollo, Real-Time Evolutionary Earthquake Location for Seismic Early Warning, Bull. seism. Soc. Am., 98:3(2008), 1482-1494.
[39] B. Schmitzer, Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems, SIAM J. Sci. Comput., 41:3(2019), A1443-A1481.
[40] M. Sharify, S. Gaubert, and L. Grigori, Solution of the optimal assignment problem by diagonal scaling algorithms, arXiv:1104.3830, 2013.
[41] P. Tong, D. Zhao and D. Yang, Tomography of the 1995 Kobe earthquake area:comparison of finite-frequency and ray approaches, Geophys. J. Int., 187(2011), 278-302.
[42] P. Tong, D. Zhao, D. Yang, X. Yang, J. Chen and Q. Liu, Wave-equation-based travel-time seismic tomography-Part 1:Method, Solid Earth, 5(2014), 1151-1168.
[43] P. Tong, D. Zhao, D. Yang, X. Yang, J. Chen and Q. Liu, Wave-equation-based travel-time seismic tomography C Part 2:Application to the 1992 Landers earthquake (Mw 7.3) area, Solid Earth, 5(2014), 1169-1188.
[44] P. Tong, D. Yang, Q. Liu, X. Yang and J. Harris, Acoustic wave-equation-based earthquake location, Geophys. J. Int., 205(2016), 464-478.
[45] C. Villani, Optimal Transport:Old and New, Springer Science & Business Media, 2008.
[46] F. Waldhauser and W.L. Ellsworth, A double-difference earthquake location algorithm:Method and application to the northern Hayward Fault, California, Bull. seism. Soc. Am., 90:6(2000), 1353-1368.
[47] X.J. Wang, On the design of a reflector antenna II, Calc. Var. Partial Dif., 20:3(2004), 329-341.
[48] Z. Wang, D. Zhou, M. Yang, Y. Zhang, C. Bao and H. Wu, Robust Document Distance with Wasserstein-Fisher-Rao Metric, in Proceedings of The 12th Asian Conference on Machine Learning, PMLR 129(2020), 721-736.
[49] X. Wen, High Order Numerical Quadratures to One Dimensional Delta Function Integrals, SIAM J. Sci. Comput., 30:4(2008), 1825-1846.
[50] H. Wu, J. Chen, X. Huang and D. Yang, A new earthquake location method based on the waveform inversion, Commun. Comput. Phys., 23:1(2018), 118-141.
[51] H. Wu and X. Yang, Eulerian Gaussian beam method for high frequency wave propagation in the reduced momentum space, Wave Motion, 50:6(2013), 1036-1049.
[52] Y. Yang, B. Engquist, J. Sun and B.D. Froese, Application of Optimal transport and the quadratic Wasserstein metric to Full-Waveform-Inversion, Geophysics, 83:1(2018), R43-R62.
[53] Y. Yang and B. Engquist, Analysis of optimal transport and related misfit functions in fullwaveform inversion, Geophysics, 83:1(2018), A7-A12.
[54] Z. Yan and Y. Wang, Full waveform inversion with sparse structure constrained regularization, J. Inverse Ill-Posed Probl., 26:2(2018), 243-257.
[55] X. Zhao, Z. Wang, Y. Zhang and H. Wu A Relaxed Matching Procedure for Unsupervised BLI, in Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, Association for Computational Linguistics, (2020), 3036-3041.
[56] W. Zhang, Acoustic multi-parameter full waveform inversion based on the wavelet method, Inverse Probl. Sci. Eng., 29:2(2021), 220-247.
[57] W. Zhang and J. Luo, Full-waveform velocity inversion based on the acoustic wave equation, Am. J. Comput. Math., 3(2013), 13-20.
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