Abstract
In a previous paper (Makropoulos andBurton, 1983) the seismic risk of the circum-Pacific belt was examined using a ‘whole process’ technique reduced to three representative parameters related to the physical release of strain energy, these are:M 1, the annual modal magnitude determined using the Gutenberg-Richter relationship;M 2, the magnitude equivalent to the total strain energy release rate per annum, andM 3, the upper bound magnitude equivalent to the maximum strain energy release in a region.
The risk analysis is extended here using the ‘part process’ statistical model of Gumbel's IIIrd asymptotic distribution of extreme values. The circum-Pacific is chosen being a complete earthquake data set, and the stability postulate on which asymptotic distributions of extremes are deduced to give similar results to those obtained from ‘whole process’ or exact distributions of extremes is successfully checked. Additionally, when Gumbel III asymptotic distribution curve fitting is compared with Gumbel I using reduced chi-squared it is seen to be preferable in all cases and it also allows extensions to an upper-bounded range of magnitude occurrences. Examining the regional seismicity generates several seismic risk results, for example, the annual mode for all regions is greater thanm(1)=7.0, with the maximum being in the Japan, Kurile, Kamchatka region atm(1)=7.6. Overall, the most hazardous areas are situated in this northwestern region and also diagonally opposite in the southeastern circum-Pacific. Relationships are established between the Gumbel III parameters and quantitiesm 1(1),X 2 and ω, quantities notionally similar toM 1,M 2 andM 3 although ω is shown to be systematically larger thanM; thereby giving a physical link through strain energy release to seismic risk statistics. Inall regions of the circum-Pacific similar results are obtained forM 1,M 2 andM 3 and the notionally corresponding statistical quantitiesm 1(1),X 2 and ω, demonstrating that the relationships obtained are valid over a wide range of seismotectonic enviroments.
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References
Aki, K. (1972),Scaling law of earthquake source time-function. Geophys. J. R. astr. Soc.,31, 3–25.
Båth, M. (1958),The energies of seismic body waves and surface waves. In:H. Benioff, M. Ewing, B. F. Howell Jr. andF. Press (Editors), Contributions in Geophysics, Pergamon, London, 1, 1–16.
Bevington, P. R. (1969),Data reduction and error analysis for the physical sciences. McGraw-Hill, New York, 336pp.
Burton, P. W. (1978a),The application of extreme value statistics to seismic hazard assessment in the European area. Proc. Symp. Anal. Seismicity and on Seismic Risk, Liblice, 17–22 October 1977. Academia, Prague 1978, 323–334.
Burton, P. W. (1978b),The IGS file of seismic activity and its use for hazard assessment. Inst. Geol. Sciences, Seism. Bul. No. 6, 13pp.
Burton, P. W. (1979),Seismic risk in Southern Europe through to India examined using Gumbel's third distribution of extreme values. Geophys. J. R. astr. Soc.,59, 249–280.
Burton, P. W. (1981),Variation in seismic risk parameters in Britain, In: Proc. 2nd Int. Symp. Anal. Seismicity and on Seismic Risk. Liblice, Czechozlovakia, 1981 May 18–23, Czechozlovak Academy of Sciences,2, 495–531.
Burton, P. W., McGonigle, R. W. Makropoulos, K. C. andUcer, S. B. (1984),Seismic risk in Turkey, the Aegean and the eastern Mediterranean: the occurrence of large magnitude earthquakes. Geophys. J. R. astr. Soc.,78, 475–506.
Cornell, C. A. andVanmarcke, E. H. (1969),The major influences on seismic risk. 4th World Conf. on Earth. Engineering, Chile.
Duda, S. J. (1965),Secular seismic energy release in the Circum-Pacific Belt. Tectonophysics,2, 409–452.
Epstein, B. andLomnitz, C. (1966),A model for the occurrence of large earthquakes. Nature,211, 954–956.
Esteva, L. (1976),Seismicity.In: Seismic Risk and Engineering Decisions, Lomnitz, C. and Rosenblueth, E., editors, Elsevier Scient. Publ. Comp., Amsterdam, 425pp.
Frechet, M. (1927),Sur la loi de probabilite de l'ecart maximum. Ann. de la Soc. polonaise de Math, Cracow,6, 93–116.
Gringorten, I. I. (1963a),Envelopes for ordered observations applied to meteorological extremes. J. Geophys. Research,68, 815–826.
Gringorten, I. I. (1963b),A plotting rule for extreme probability paper. J. Geophys. Research,68, 813–814.
Gumbel, E. J. (1935),Les valeurs extrèmes des distribution statistiques, Ann. Inst. Henri Poincaré,5, 115–158.
Gumbel, E. J. (1967),Statistics of extremes. Columbia Univ. Press, New York, 375 pp.
Gutenberg, B. (1956),Great earthquakes 1896–1903. Trans. Am. Geophys. Union,37, 608–614.
Gutenberg, B. andRichter, C. F. (1944)Frequency of earthquakes in California. Bull. Seism. Soc. Am.,34, 185–188.
Gutenberg, B. andRichter, C. F. (1954),Seismicity of the earth and associated phenomena. Princeton Univ. Press, Princeton, N. J., 310pp.
Jenkinson, A. F. (1955),The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q. Jour. Roy. Meteor. Soc.,87, 158–171.
Kanamori, H. (1977),The energy release in great earthquakes. J. Geophys. Res.,82, 2981–2987.
Kanamori, H. (1978),Quantification of earthquakes. Nature,271, 411–414.
Karnik, V. andHubnerova, Z., (1968),The probability of occurrence of largest earthquakes in the European area. Pure appl. Geophys.,70, 61–73.
Krumbein, W. C. andLieblein, J. (1956),Geological application of extreme value methods to interpretation of cobbles and boulders in gravel deposits. Trans. Amer. Geophys. Union,37, 313–319.
Levenburg, K. (1944),A method for the solution of certain non-linear problems in least squares. Q. Appl. Math.,2, 164–168.
Lomnitz, C. (1974),Global tectonics and earthquake risk. Elsev, Scient. Publ. Comp. Amsterdam, 320pp.
Makropoulos, K. C. (1978),The statistics of large earthquake magnitude and an evaluation of Greek seismicity. Ph.D. thesis, University of Edinburgh.
Makropoulos, K. C. andBurton, P. W. (1983),Seismic risk of circum-Pacific earthquakes: I Strain energy release. Pure appl. Geophys.,121, 247–267.
Marquardt, D. W. (1963),An algorithm for least-squares estimation of nonlinear parameters J. Soc. Indust. App. Maths.,11, 431–441.
Merz, H. andCornell, C. A. (1973),Seismic risk analysis based on a quadratic magnitude-frequency law. Bull. Seim. Soc. Am.,63, 1999–2006.
Nordquist, J. M. (1945),Theory of large values applied to earthquake magnitudes. Trans. Amer. Geophys. Union,26, 29–31.
Radu, C. andApopei, G. (1977),Application of the largest values theory to Vrancea earthquakes. Inst. Geophys., Polish Acad. Sci.,A5 (116), 229–243.
Richter, C. F. (1958),Elementary Seismology, Freeman and Co., San Francisco, 766pp.
Roca, A., Arroyo, A. L. andSurinagh, E. (1984),Application of the Gumbel III law to seismic data from southern Spain. Eng. Geol.,20, 63–71.
Sacuiu, I. andZorilescu, D. (1970),Statistical analysis of seismic data on earthquakes in the area of the Vrancea focus. Bull. Seism. Soc. Am.,60, 1089–1099.
Schenkova, Z. andKarnik, V. (1970),The probability of occurrence of largest earthquakes in the European areas—Part II. Pure appl. Geophys.,80, 152–161.
Schenkova, Z. andKarnik, V. (1977),Statistical prediction of the maximum magnitude earthquakes in the Balkan region. Inst. Geophys., Polish Acad. Sci.,A-5 (116) 245–250.
Schenkova, Z. andKarnik, V. (1978),The third asymptotic distribution of largest magnitudes in the Balkan earthquake provinces. Pure appl. Geophys.,116, 1314–1325.
Watson, G. S. (1954),Extreme values in samples from m-dependent stationary stochastic processes. Ann. Math. Stats.,25, 798–800.
Yegulalp, T. M., andKuo, J. T. (1974),Statistical prediction of the occurrence of maximum magnitude earthquakes. Bull. Seism. Soc. Am.,64, 393–414.
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Burton, P.W., Makropoulos, K.C. Seismic risk of circum-Pacific earthquakes: II. Extreme values using Gumbel's third distribution and the relationship with strain energy release. PAGEOPH 123, 849–869 (1985). https://doi.org/10.1007/BF00876974
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DOI: https://doi.org/10.1007/BF00876974