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Estimating Earthquake-Induced Tsunami Height Probabilities without Sampling

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Abstract

Given a distribution of earthquake-induced seafloor elevations, we present a method to compute the probability of the resulting tsunamis reaching a certain size on shore. Instead of sampling, the proposed method relies on optimization to compute the most likely fault slips that result in a seafloor deformation inducing a large tsunami wave. We model tsunamis induced by bathymetry change using the shallow water equations on an idealized slice through the sea. The earthquake slip model is based on a sum of multivariate log-normal distributions, and follows the Gutenberg-Richter law for seismic moment magnitudes ranging from 7 to 9. For a model problem inspired by the Tohoku-Oki 2011 earthquake and tsunami, we quantify annual probabilities of differently sized tsunami waves. Our method also identifies the most effective tsunami mechanisms. These mechanisms have smoothly varying fault slip patches that lead to an expansive but moderately large bathymetry change. The resulting tsunami waves are compressed as they approach shore and reach close-to-vertical leading wave edge close to shore.

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References

  • An, C., Sepúlveda, I., & Liu, P.L.-F. (2014). Tsunami source and its validation of the 2014 Iquique, Chile, earthquake. Geophysical Research Letters, 41(11), 3988–3994.

    Article  Google Scholar 

  • Behrens, J., Løvholt, F., Jalayer, F., Lorito, S., Salgado-Gálvez, M. A., Sørensen, M., Abadie, S., Aguirre-Ayerbe, I., Aniel-Quiroga, I., Babeyko, A., et al. (2021). Probabilistic tsunami hazard and risk analysis: A review of research gaps. Frontiers in Earth Science, 9, 628772.

    Article  Google Scholar 

  • Borzi, A., & Schulz, V. (2011). Computational optimization of systems governed by partial differential equations (Vol. 8). SIAM.

    Book  Google Scholar 

  • Crempien, J. G. F., Urrutia, A., Benavente, R., & Cienfuegos, R. (2020). Effects of earthquake spatial slip correlation on variability of tsunami potential energy and intensities. Scientific Reports, 10(1), 1–10.

    Article  Google Scholar 

  • Dao, M. H., & Tkalich, P. (2007). Tsunami propagation—A sensitivity study. Natural Hazards and Earth System Sciences, 7(6), 741–754.

    Article  Google Scholar 

  • De Los Reyes, J. C. (2015). Numerical PDE-constrained optimization. Springer.

    Book  Google Scholar 

  • Dematteis, G., Grafke, T., & Vanden-Eijnden, E. (2018). Rogue waves and large deviations in deep sea. Proceedings of the National Academy of Sciences, 115(5), 855–860.

    Article  Google Scholar 

  • Dematteis, G., Grafke, T., & Vanden-Eijnden, E. (2019). Extreme event quantification in dynamical systems with random components. SIAM/ASA Journal on Uncertainty Quantification, 7(3), 1029–1059.

    Article  Google Scholar 

  • Fujiwara, T., Kodaira, S., No, T., Kaiho, Y., Takahashi, N., & Kaneda, Y. (2011). The 2011 Tohoku-Oki earthquake: Displacement reaching the trench axis. Science, 334(6060), 1240–1240. https://doi.org/10.1126/science.1211554

    Article  Google Scholar 

  • Gao, D., Wang, K., Insua, T. L., Sypus, M., Riedel, M., & Sun, T. (2018). Defining megathrust tsunami source scenarios for northernmost Cascadia. Natural Hazards, 94(1), 445–469.

    Article  Google Scholar 

  • Grezio, A., Babeyko, A., Baptista, M. A., Behrens, J., Costa, A., Davies, G., Geist, E. L., Glimsdal, S., González, F. I., Griffin, J., et al. (2017). Probabilistic tsunami hazard analysis: Multiple sources and global applications. Reviews of Geophysics, 55(4), 1158–1198.

    Article  Google Scholar 

  • Halko, N., Martinsson, P. G., & Tropp, J. A. (2011). Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions. SIAM Review, 53(2), 217–288.

    Article  Google Scholar 

  • Hashima, A., Becker, T. W., Freed, A. M., Sato, H., & Okaya, D. A. (2016). Coseismic deformation due to the 2011 Tohoku-oki earthquake: Influence of 3-D elastic structure around Japan. Earth, Planets and Space, 68(1), 1–15.

    Article  Google Scholar 

  • Hesthaven, J. S., & Warburton, T. (2007). Nodal discontinuous Galerkin methods: Algorithms, analysis, and applications. Springer.

    Google Scholar 

  • Hinze, M., Pinnau, R., Ulbrich, M., & Ulbrich, S. (2009). Optimization with PDE constraints. Springer.

    Google Scholar 

  • Kagan, Y. Y., & Jackson, D. D. (2013). Tohoku earthquake: A surprise? Bulletin of the Seismological Society of America, 103(2B), 1181–1194.

    Article  Google Scholar 

  • Lavallée, D., & Archuleta, R. J. (2003). Stochastic modeling of slip spatial complexities for the 1979 Imperial Valley, California, earthquake. Geophysical Research Letters, 30(5), 622–640.

    Article  Google Scholar 

  • Lavallée, D., Liu, P., & Archuleta, R. J. (2006). Stochastic model of heterogeneity in earthquake slip spatial distributions. Geophysical Journal International, 165(2), 622–640.

    Article  Google Scholar 

  • LeVeque, R. J., Waagan, K., González, F. I., Rim, D. & Lin, G. (2016) Generating random earthquake events for probabilistic tsunami hazard assessment. In Global Tsunami Science: Past and Future (vol. I, pp. 3671–3692). Cham: Birkhäuser.

  • Liu, J. S. (2001). Monte Carlo strategies in scientific computing (Vol. 10). Springer.

    Google Scholar 

  • Mai, P. M., & Beroza, G. C. (2002). A spatial random field model to characterize complexity in earthquake slip. Journal of Geophysical Research: Solid Earth, 107(B11), 10.

    Article  Google Scholar 

  • Murotani, S., Miyake, H., & Koketsu, K. (2008). Scaling of characterized slip models for plate-boundary earthquakes. Earth, Planets and Space, 60(9), 987–991.

    Article  Google Scholar 

  • Nocedal, J., & Wright, S. J. (2006). Numerical optimization (2nd ed.). Springer.

    Google Scholar 

  • Okada, Y. (1985). Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 75(4), 1135–1154.

    Article  Google Scholar 

  • Rackwitz, R. (2001). Reliability analysis—A review and some perspectives. Structural Safety, 23(4), 365–395.

    Article  Google Scholar 

  • Tong, S., Vanden-Eijnden, E., & Stadler, G. (2021). Extreme event probability estimation using PDE-constrained optimization and large deviation theory, with application to tsunamis. Communications in Applied Mathematics and Computational Science, 16(2), 181–225.

    Article  Google Scholar 

  • Williamson, A. L., Rim, D., Adams, L. M., LeVeque, R. J., Melgar, D., & González, F. I. (2020). A source clustering approach for efficient inundation modeling and regional scale probabilistic tsunami hazard assessment. Frontiers in Earth Science, 8, 591663.

  • Zhan, Z., Helmberger, D., Simons, M., Kanamori, H., Wu, W., Cubas, N., Duputel, Z., Chu, R., Tsai, V. C., Avouac, J.-P., et al. (2012). Anomalously steep dips of earthquakes in the 2011 Tohoku-Oki source region and possible explanations. Earth and Planetary Science Letters, 353, 121–133.

    Article  Google Scholar 

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Funding

S. T. and G. S. were partially supported by the US National Science Foundation (NSF) through Grant DMS #1723211. E. V.-E. was supported in part by the NSF Materials Research Science and Engineering Center Program Grant DMR #1420073, by NSF Grant DMS #152276, by the Simons Collaboration on Wave Turbulence, Grant #617006, and by ONR Grant #N4551-NV-ONR.

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Authors and Affiliations

  1. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    Shanyin Tong

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Shanyin Tong, Eric Vanden-Eijnden & Georg Stadler

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  1. Shanyin Tong
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  2. Eric Vanden-Eijnden
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  3. Georg Stadler
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Correspondence to Shanyin Tong.

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Tong, S., Vanden-Eijnden, E. & Stadler, G. Estimating Earthquake-Induced Tsunami Height Probabilities without Sampling. Pure Appl. Geophys. 180, 1587–1597 (2023). https://doi.org/10.1007/s00024-023-03281-3

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