Discrete scale invariance, complex fractal dimensions, and log‐periodic fluctuations in seismicity

We discuss in detail the concept of discrete scale invariance and show how it leads to complex critical exponents and hence to the log-periodic corrections to scaling exhibited by various measures of seismic activity close to a large earthquake singularity. Discrete scale invariance is first illustrated on a geometrical fractal, the Sierpinsky gasket, which is shown to be fully described by a complex fractal dimension whose imaginary part is a simple function (inverse of the logarithm) of the discrete scaling factor. Then, a set of simple physical systems (spins and percolation) on hierarchical lattices is analyzed to exemplify the origin of the different terms in the discrete renormalization group formalism introduced to tackle this problem. As a more specific example of rupture relevant for earthquakes, we propose a solution of the hierarchical time-dependent fiber bundle of Newman et al. [1994] which exhibits explicitly a discrete renormalization group from which log-periodic corrections follow. We end by pointing out that discrete scale invariance does not necessarily require an underlying geometrical hierarchical structure. A hierarchy may appear “spontaneously” from the physics and/or the dynamics in a Euclidean (nonhierarchical) heterogeneous system. We briefly discuss a simple dynamical model of such mechanism, in terms of a random walk (or diffusion) of the seismic energy in a random heterogeneous system.

[1]  Sornette,et al.  Dynamics and memory effects in rupture of thermal fuse networks. , 1992, Physical review letters.

[2]  Henry,et al.  First observation of the universal periodic corrections to scaling: Magnetoresistance of normal-metal self-similar networks. , 1986, Physical review letters.

[3]  Stuart P. Nishenko,et al.  Seismicity trends and potential for large earthquakes in the Alaska-Aleutian region , 1994 .

[4]  Michael Ghil,et al.  Deep water formation and Quaternary glaciations , 1987 .

[5]  D. Sornette,et al.  Discrete Scaling in Earthquake Precursory Phenomena: Evidence in the Kobe Earthquake, Japan , 1996 .

[6]  Allan G. Lindh.  The seismic cycle pursued , 1990, Nature.

[7]  Didier Sornette,et al.  Complex Critical Exponents from Renormalization Group Theory of Earthquakes: Implications for Earthquake Predictions , 1995 .

[8]  David J. Varnes,et al.  Predictive modeling of the seismic cycle of the Greater San Francisco Bay Region , 1993 .

[9]  J. Eaton.  The earthquake and its aftershocks from May 2 through September 30, 1983 , 1990 .

[10]  Yan Y. Kagan,et al.  Statistical study of the occurrence of shallow earthquakes , 1978 .

[11]  J. Bernasconi,et al.  Diffusion in a one-dimensional lattice with random asymmetric transition rates , 1982 .

[12]  John B. Rundle,et al.  A physical model for earthquakes: 3. Thermodynamical approach and its relation to nonclassical theories of nucleation , 1989 .

[13]  K. Binder,et al.  Spin glasses: Experimental facts, theoretical concepts, and open questions , 1986 .

[14]  Frederick Solomon.  Random Walks in a Random Environment , 1975 .

[15]  West,et al.  Complex fractal dimension of the bronchial tree. , 1991, Physical review letters.

[16]  J. Rundle.  A physical model for earthquakes: 2. Application to southern California , 1988 .

[17]  D. Turcotte,et al.  An exact renormalization model for earthquakes and material failure: statics and dynamics , 1994 .

[18]  D. J. Wallace,et al.  Gradient properties of the renormalisation group equations in multicomponent systems , 1975 .

[19]  A. Provost,et al.  Scaling rules in rock fracture and possible implications for earthquake prediction , 1982, Nature.

[20]  Tamaz Chelidze,et al.  Percolation and fracture , 1982 .

[21]  C. Scholz.  The Mechanics of Earthquakes and Faulting , 1990 .

[22]  G. King,et al.  The mechanisms of finite brittle strain , 1992 .

[23]  B. Derrida,et al.  Renormalisation groups with periodic and aperiodic orbits , 1983 .

[24]  D. Vere-Jones,et al.  Statistical theories of crack propagation , 1977 .

[25]  D. Sornette,et al.  Earthquake rupture as a critical point: consequences for telluric precursors , 1990 .

[26]  Sara A. Solla,et al.  A renormalization group approach to the stick-slip behavior of faults , 1985 .

[27]  Knopoff,et al.  Avalanches and epidemic models of fracturing in earthquakes. , 1992, Physical Review A. Atomic, Molecular, and Optical Physics.

[28]  Sammis,et al.  Fractal distribution of earthquake hypocenters and its relation to fault patterns and percolation. , 1993, Physical review letters.

[29]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[30]  Lucile M. Jones,et al.  Foreshocks, aftershocks, and earthquake probabilities: Accounting for the landers earthquake , 1994 .

[31]  Bernard D. Coleman,et al.  A Stochastic Process Model for Mechanical Breakdown , 1957 .

[32]  G. Jona-Lasinio.  The renormalization group: A probabilistic view , 1975 .

[33]  M. Nauenberg.  Scaling representation for critical phenomena , 1975 .

[34]  T. Lubensky,et al.  Mean field and ε -expansion study of spin glasses , 1977 .

[35]  D. Varnes,et al.  The cyclic and fractal seismic series preceding an mb 4.8 earthquake on 1980 February 14 near the Virgin Islands , 1996 .

[36]  S. L. Phoenix,et al.  A statistical model for the time dependent failure of unidirectional composite materials under local elastic load-sharing among fibers , 1983 .

[37]  Vladimir Keilis-Borok,et al.  Introduction: Non-linear systems in the problem of earthquake prediction , 1990 .

[38]  Clément Narteau,et al.  Scaling organization of fracture tectonics (SOFT) and earthquake mechanism , 1995 .

[39]  Ronald L. Biegel,et al.  The kinematics of gouge deformation , 1987 .

[40]  Sornette,et al.  Statistical model of earthquake foreshocks. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[41]  J. M. J. van Leeuwen,et al.  Real-Space Renormalization , 1982 .

[42]  Michael Ghil,et al.  Boolean delay equations. II. Periodic and aperiodic solutions , 1985 .

[43]  H. Herrmann,et al.  Statistical models for the fracture of disordered media. North‐Holland, 1990, 353 p., ISBN 0444 88551x (hardbound) US $ 92.25, 0444 885501 (paperback) US $ 41.00 , 1990 .

[44]  C. R. Allen,et al.  Intermediate-term prediction of occurrence times of strong earthquakes , 1988, Nature.

[45]  F. Y. Wu.  The Potts model , 1982 .

[46]  A. Aharony.  Critical properties of random and constrained dipolar magnets , 1975 .

[47]  Lynn R. Sykes,et al.  Seismic activity on neighbouring faults as a long-term precursor to large earthquakes in the San Francisco Bay area , 1990, Nature.

[48]  B. Voight,et al.  A Relation to Describe Rate-Dependent Material Failure , 1989, Science.

[49]  E. Novikov.  The effects of intermittency on statistical characteristics of turbulence and scale similarity of breakdown coefficients , 1990 .

[50]  D. L. Anderson,et al.  Forecasting Southern California Earthquakes , 1982, Science.

[51]  K. Wilson.  Problems in Physics with many Scales of Length , 1979 .

[52]  Didier Sornette,et al.  Linking short-timescale deformation to long-timescale tectonics , 1992, Nature.

[53]  Michael Ghil,et al.  Boolean difference equations. I - Formulation and dynamic behavior , 1984 .

[54]  Claude J. Allègre,et al.  Introduction of scaling techniques in brittle fracture of rocks , 1994 .

[55]  J. Geronimo,et al.  Complex spectral dimensionality on fractal structures , 1983 .

[56]  D. P. Schwartz,et al.  Fault behavior and characteristic earthquakes: Examples from the Wasatch and San Andreas Fault Zones , 1984 .

[57]  John B. Rundle,et al.  A physical model for earthquakes: 1. Fluctuations and interactions , 1988 .

[58]  F. Anselmet,et al.  High-order velocity structure functions in turbulent shear flows , 1984, Journal of Fluid Mechanics.