A fractal approach to probabilistic seismic hazard assessment

Abstract The definition of a fractal distribution is that the number of objects (events) N with a characteristic size greater than r satisfies the relation N ~ r −D where D is the fractal dimension. The applicability of a fractal relation implies that the underlying physical process is scale invariant over the range of applicability of the relation. The empirical frequency-magnitude relation for earthquakes defining a b- value is a fractal relation with D = 2b . Accepting the fractal distribution, the level of regional seismicity can be related to the rate of regional strain and the magnitude of the largest characteristic earthquake. High levels of seismic activity indicate either a large regional strain or a low-magnitude maximum characteristic earthquake (or both). If the regional seismicity has a weak time dependence, the approach can be used to make probabilistic seismic hazard assessments. As in many fractal applications, the basic approach has been used in the past. However, the association of a fractal behavior with scale invariance provides a rational basis for the extrapolation of frequency-magnitude statistics in order to make predictions of seismic hazards.

[1]  John G. Anderson.  Seismic strain rates in the Central and Eastern United States , 1986 .

[2]  D. L. Turcotte,et al.  A fractal model for crustal deformation , 1986 .

[3]  P. Grassberger.  On the Hausdorff dimension of fractal attractors , 1981 .

[4]  Peter Molnar,et al.  Earthquake recurrence intervals and plate tectonics , 1979 .

[5]  Donald L. Turcotte,et al.  A fractal approach to the relationship between ore grade and tonnage , 1986 .

[6]  Hiroo Kanamori,et al.  Quantification of Earthquakes , 1978, Nature.

[7]  P. Burton,et al.  LONG-TERM EARTHQUAKE RECURRENCE CONSTRAINED BY TECTONIC SEISMIC MOMENT RELEASE RATES , 1986 .

[8]  J. Enrique Luco,et al.  Consequences of slip rate constraints on earthquake occurrence relations , 1983 .

[9]  Donald L. Turcotte,et al.  Fractals and fragmentation , 1986 .

[10]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[11]  Kerry E Sieh,et al.  Prehistoric large earthquakes produced by slip on the San Andreas Fault at Pallett Creek, California , 1978 .

[12]  H. Kanamori,et al.  A moment magnitude scale , 1979 .

[13]  Richard H. Jahns,et al.  Holocene activity of the San Andreas fault at Wallace Creek , 1984 .

[14]  Geoffrey C. P. King,et al.  The accommodation of large strains in the upper lithosphere of the earth and other solids by self-similar fault systems: the geometrical origin of b-Value , 1983 .

[15]  A. Ruina.  Slip instability and state variable friction laws , 1983 .

[16]  R. Hyndman,et al.  Seismicity and rates of relative motion on the plate boundaries of Western North America , 1983 .

[17]  D. L. Anderson,et al.  Theoretical Basis of Some Empirical Relations in Seismology by Hiroo Kanamori And , 1975 .

[18]  B. Gutenberg,et al.  Seismicity of the Earth and associated phenomena , 1950, MAUSAM.

[19]  Keiiti Aki,et al.  Magnitude‐frequency relation for small earthquakes: A clue to the origin of ƒmax of large earthquakes , 1987 .

[20]  Thomas V. McEvilly,et al.  Recurrence models and Parkfield, California, earthquakes , 1984 .

[21]  E. Ott,et al.  Dimension of Strange Attractors , 1980 .

[22]  Robert R. Youngs,et al.  Implications of fault slip rates and earthquake recurrence models to probabilistic seismic hazard estimates , 1985 .

[23]  John G. Anderson.  Estimating the seismicity from geological structure for seismic-risk studies , 1979 .

[24]  J. Evernden,et al.  Study of regional seismicity and associated problems , 1970, Bulletin of the Seismological Society of America.

[25]  Luis Esteva,et al.  Statistics of small earthquakes and frequency of occurrence of large earthquakes along the Mexican subduction zone , 1983 .

[26]  B. Mandelbrot.  How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension , 1967, Science.

[27]  Stewart w. Smith,et al.  Determination of maximum earthquake magnitude , 1976 .