Reconstruction of Far-Field Tsunami Amplitude Distributions from Earthquake Sources

  • Eric L. GeistEmail author
  • Tom Parsons
Part of the Pageoph Topical Volumes book series (PTV)


The probability distribution of far-field tsunami amplitudes is explained in relation to the distribution of seismic moment at subduction zones. Tsunami amplitude distributions at tide gauge stations follow a similar functional form, well described by a tapered Pareto distribution that is parameterized by a power-law exponent and a corner amplitude. Distribution parameters are first established for eight tide gauge stations in the Pacific, using maximum likelihood estimation. A procedure is then developed to reconstruct the tsunami amplitude distribution that consists of four steps: (1) define the distribution of seismic moment at subduction zones; (2) establish a source-station scaling relation from regression analysis; (3) transform the seismic moment distribution to a tsunami amplitude distribution for each subduction zone; and (4) mix the transformed distribution for all subduction zones to an aggregate tsunami amplitude distribution specific to the tide gauge station. The tsunami amplitude distribution is adequately reconstructed for four tide gauge stations using globally constant seismic moment distribution parameters established in previous studies. In comparisons to empirical tsunami amplitude distributions from maximum likelihood estimation, the reconstructed distributions consistently exhibit higher corner amplitude values, implying that in most cases, the empirical catalogs are too short to include the largest amplitudes. Because the reconstructed distribution is based on a catalog of earthquakes that is much larger than the tsunami catalog, it is less susceptible to the effects of record-breaking events and more indicative of the actual distribution of tsunami amplitudes.


Tsunamis probability distribution seismic moment tsunami amplitude tide gauge 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abe K. (1979), Size of great earthquake of 1837–1974 inferred from tsunami data, J. Geophys. Res., 84, 1561–1568.CrossRefGoogle Scholar
  2. Abe K. (1989), Quanitification of tsunamigenic earthquakes by the Mt scale, Tectonophys., 166, 27–34.Google Scholar
  3. Ben-Menahem A., Rosenman M. (1972), Amplitude patterns of tsunami waves from submarine earthquakes, J. Geophys. Res., 77, 3097–3128.CrossRefGoogle Scholar
  4. Bird P., Kagan Y.Y. (2004), Plate-tectonic analysis of shallow seismicity: apparent boundary width, beta-value, corner magnitude, coupled lithosphere thickness, and coupling in 7 tectonic settings, Bull. Seismol. Soc. Am., 94, 2380–2399.Google Scholar
  5. Burroughs S.M., Tebbens S.F. (2001), Upper-truncated power laws in natural systems, Pure Appl. Geophys., 158, 741–757.CrossRefGoogle Scholar
  6. Burroughs S.M., Tebbens S.F. (2005), Power law scaling and probabilistic forecasting of tsunami runup heights, Pure Appl. Geophys., 162, 331–342.CrossRefGoogle Scholar
  7. Clauset A., Shalizi C.R., Newman M.E.J. (2009), Power-law distributions in empirical data, SIAM Review, 51, 661–703.CrossRefGoogle Scholar
  8. Comer R.P. (1980), Tsunami height and earthquake magnitude: theoretical basis of an empirical relation, Geophys. Res. Lett., 7, 445–448.CrossRefGoogle Scholar
  9. Ekström G., Nettles M. (1997), Calibration of the HGLP seismograph network and centroid-moment tensor analysis of significant earthquakes of 1976, Physics of the Earth and Planetary Interiors, 101, 221–246.CrossRefGoogle Scholar
  10. Engdahl E.R., Villaseñor A. (2002), Global seismicity: 1900-1999. In: Lee WHK, Kanamori H, Jennings PC, Kisslinger C (eds), International Handbook of Earthquake and Engineering Seismology, Part A. Academic Press, San Diego, pp. 665–690.Google Scholar
  11. Geist E.L. (1999), Local tsunamis and earthquake source parameters, Adv. Geophys., 39, 117–209.Google Scholar
  12. Geist E.L. (2012), Phenomenology of tsunamis II: scaling, Event Statistics, and Inter-Event Triggering, Adv. Geophys., 53, 35–92.Google Scholar
  13. Geist E.L. (2014), Explanation of temporal clustering of tsunami sources using the epidemic-type aftershock sequence model, Bull. Seismol. Soc. Am., 104, 2091–2103.CrossRefGoogle Scholar
  14. Geist E.L., Parsons T. (2006), Probabilistic analysis of tsunami hazards, Natural Hazards, 37, 277–314.CrossRefGoogle Scholar
  15. Geist E.L., Parsons T. (2011), Assessing historical rate changes in global tsunami occurrence, Geophys. J. Int., 187, 497–509.Google Scholar
  16. Geist E.L., Parsons T. (2014), Undersampling power-law size distributions: effect on the assessment of extreme natural hazards, Natural Hazards, 72, 565-595. doi: 10.1007/s11069-013-1024-0.CrossRefGoogle Scholar
  17. Geist E.L., Parsons T., ten Brink U.S., Lee H.J. (2009), Tsunami Probability. In: Bernard EN, Robinson AR (eds), The Sea, v. 15. Harvard University Press, Cambridge, Massachusetts, pp. 93–135.Google Scholar
  18. Geist E.L., ten Brink U.S., Gove M. (2014), A framework for the probabilistic analysis of meteotsunamis, Natural Hazards, 74, 123-142. doi: 10.1007/s11069-014-1294-1.CrossRefGoogle Scholar
  19. Geller R.J., Kanamori H. (1977), Magnitudes of great shallow earthquakes from 1904 to 1952, Bull. Seismol. Soc. Am., 67, 587–598.Google Scholar
  20. Gutenberg B., Richter C.F. (1944), Frequency of earthquakes in California, Bull. Seismol. Soc. Am., 34, 185–188.Google Scholar
  21. Hatori T. (1971), Tsunami sources in Hokkaido and southern Kuril regions, Bulletin of the Earthquake Research Institute, 49, 63–75.Google Scholar
  22. Horrillo J., Knight W., Kowalik Z. (2008), Kuril Islands tsunami of November 2006: 2. Impact at Crescent City by local enhancement, J. Geophys. Res., 113, doi: 10.1029/2007JC004404.
  23. Huber P.J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In: Proceedings of the fifth Berkeley symposium on mathematica statistics and probability, pp. 221–233.Google Scholar
  24. Ishimoto M., Iida K. (1939), Observations of earthquakes registered with the microseismograph constructed recently, Bulletin of the Earthquake Research Institute, 17, 443–478.Google Scholar
  25. Kagan Y.Y. (1997), Seismic moment-frequency relation for shallow earthquakes: regional comparison, J. Geophys. Res., 102, 2835–2852.CrossRefGoogle Scholar
  26. Kagan Y.Y. (1999), Universality of the seismic-moment-frequency relation, Pure Appl. Geophys., 155, 537–573.CrossRefGoogle Scholar
  27. Kagan Y.Y. (2002a), Seismic moment distribution revisited: I. Statistical results, Geophys. J. Int., 148, 520–541.CrossRefGoogle Scholar
  28. Kagan Y.Y. (2002b), Seismic moment distribution revisited: II. Moment conservation principle, Geophys. J. Int., 149, 731–754.CrossRefGoogle Scholar
  29. Kagan Y.Y. (2010), Earthquake size distribution: power-law with exponent β = 1/2?, Tectonophys., 490, 103–114.Google Scholar
  30. Kagan Y.Y., Bird P., Jackson D.D. (2010), Earthquake patterns in diverse tectonic zones of the globe, Pure Appl. Geophys., 167, 721–741.CrossRefGoogle Scholar
  31. Kagan Y.Y., Jackson D.D. (2013), Tohoku earthquake: a surprise?, Bull. Seismol. Soc. Am., 103, 1181–1194.CrossRefGoogle Scholar
  32. Kempthorne O., Folks L. (1971), Probability, statistics, and data analysis. Iowa State University Press, Ames, Iowa.Google Scholar
  33. López-Ruiz R., Vázquez-Prada M., Gómez J.B., Pacheco A.F. (2004), A model of characteristic earthquakes and its implications for regional seismicity, Terra Nova, 16, 116–120.CrossRefGoogle Scholar
  34. Main I., Naylor M., Greenhough J., Touati S., Bell A.F., McCloskey J. (2011), Model selection and uncertainty in earthquake hazard analysis. In: Faber M, Köhler J, Nishijima K (eds), Applications of Statistics and Probability in Civil Engineering. CRC Press, Leiden, The Netherlands, pp. 735–743.CrossRefGoogle Scholar
  35. McCaffrey R. (2008), Global frequency of magnitude 9 earthquakes, Geology, 36, 263–266.CrossRefGoogle Scholar
  36. Okal E.A. (1988), Seismic parameters controlling far-field tsunami amplitudes: a review, Natural Hazards, 1, 67–96.Google Scholar
  37. Olami Z., Feder H.J.S., Christensen K. (1992), Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Physical Review Letters, 68, 1244–1247.CrossRefGoogle Scholar
  38. Pacheco J.F., Sykes L.R. (1992), Seismic moment catalog of large shallow earthquakes, 1900 to 1989, Bull. Seismol. Soc. Am., 82, 1306–1349.Google Scholar
  39. Parsons T., Console R., Falcone G., Murru M., Yamashina K. (2012), Comparison of characteristic and Gutenberg-Richter models for time-dependent M  7.9 earthquake probability in the Nankai-Tokai subduction zone, Japan, Geophys. J. Int., doi: 10.1111/j.1365-1246X.2012.05595.x.
  40. Parsons T., Geist E.L. (2009), Is there a basis for preferring characteristic earthquakes over a Gutenberg-Richter distribution in probabilistic earthquake forecasting?, Bull. Seismol. Soc. Am., 99, 2012–2019. doi: 10.1785/0120080069.CrossRefGoogle Scholar
  41. Parsons T., Geist E.L. (2012), Were global M  8.3 earthquake time intervals random between 1900–2011?, Bull. Seismol. Soc. Am., 102, doi: 10.1785/0120110282.CrossRefGoogle Scholar
  42. Parsons T., Geist E.L. (2014), The 2010–2014.3 global earthquake rate increase, Geophys. Res. Lett., 41, 4479–4485. doi: 10.1002/2014GL060513.CrossRefGoogle Scholar
  43. Pawitan Y. (2001), In all likelihood: statistical modelling and inference using likelihood. Oxford University Press, Oxford.Google Scholar
  44. Pelayo A.M., Wiens D.A. (1992), Tsunami earthquakes: slow thrust-faulting events in the accretionary wedge, J. Geophys. Res., 97, 15,321–315,337.CrossRefGoogle Scholar
  45. Rabinovich A.B., Thomson R.E. (2007), The 26 December 2004 Sumatra tsunami: analysis of tide gauge data from the world ocean Part 1. Indian Ocean and South Africa, Pure Appl. Geophys., 164, 261–308.CrossRefGoogle Scholar
  46. Satake K., Okada M., Abe I. (1988), Tide gauge response to tsunamis: measurements at 40 tide gauge stations in Japan, Journal of Marine Research, 46, 557–571.CrossRefGoogle Scholar
  47. Sornette D. (2009), Probability distribution in complex systems. In: Meyers RA (ed), Encyclopedia of Complexity and Systems Science. Springer, New York, pp. 7009–7024.CrossRefGoogle Scholar
  48. Vere-Jones D., Robinson R., Yang W. (2001), Remarks on the accelerated moment release model: problems of model formulation, simulation and estimation, Geophys. J. Int., 144, 517–531.CrossRefGoogle Scholar
  49. Wesnousky S.G. (1994), The Gutenberg-Richter or characteristic earthquake distribution, which is it?, Bull. Seismol. Soc. Am., 84, 1940–1959.Google Scholar
  50. White H. (1980), A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, Econometrica, 48, 817–838.CrossRefGoogle Scholar
  51. Zöller G. (2013), Convergence of the frequency-magnitude distribution of global earthquakes: maybe in 200 years, Geophys. Res. Lett., 40, 3873–3877.CrossRefGoogle Scholar

Copyright information

© Springer (outside the USA) 2016

Authors and Affiliations

  1. 1.U.S. Geological SurveyMenlo ParkUSA

Personalised recommendations