Contents

## Yan Y. Kagan Earthquakes in addition to Fractures in Solids: Why do we fail to underst in addition to them in addition to what can be done Outline Two Major Unsolved Problems of Modern Science

Devine, Caribe, Meteorologist has reference to this Academic Journal, PHwiki organized this Journal Yan Y. Kagan Dept. Earth in addition to Space Sciences, UCLA, Los Angeles, CA 90095-1567, ykagan@ucla.edu, http://scec.ess.ucla.edu/ykagan.html Earthquakes in addition to Fractures in Solids: Why do we fail to underst in addition to them in addition to what can be done http://scec.ess.ucla.edu/~ykagan/india-index.html Outline 1. Fracture in addition to turbulence – no significant theoretical progress. 2. Deficiencies of present physical models as long as earthquake occurrence. 3. Phenomenology: fractal distributions of size, time, space, in addition to focal mechanisms. 4. Fractal model of earthquake process: r in addition to om stress interactions. 5. Statistical as long as ecasting earthquakes in addition to its testing (more tomorrow at 12:00 in room 1707). Two Major Unsolved Problems of Modern Science 1. Turbulent flow of fluids (Navier-Stocks equations). 2. Brittle fracture of solids. Plastic de as long as mation of materials is an intermediate case: it behaves as a solid as long as short-term interaction in addition to as a liquid as long as long-term interaction. Kagan, Y. Y., 1992. Seismicity: Turbulence of solids, Nonlinear Science Today, 2, 1-13.

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Navier-Stokes Equation Waves follow our boat as we me in addition to er across the lake, in addition to turbulent air currents follow our flight in a modern jet. Mathematicians in addition to physicists believe that an explanation as long as in addition to the prediction of both the breeze in addition to the turbulence can be found through an underst in addition to ing of solutions to the Navier-Stokes equations. Although these equations were written down in the 19-th Century, our underst in addition to ing of them remains minimal. The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations (Clay Institute – one of seven math millennium problems – prize $1,000,000). Akiva Yaglom (2001, p. 4) commented that the turbulence status is different from many other complex problems that 20-th century physics solved or was trying to solve: “However, turbulence theory deals with the most ordinary in addition to simple realities of the everyday life such as, e.g., the jet of water spurting from the kitchen tap.” Nevertheless, the turbulence problem is not among the ten millennium problems in physics presented by University of Michigan Ann Arbor, see http://feynman.physics.lsa.umich.edu/strings2000/millennium.html or 11 problems by the National Research Council’s board on physics in addition to astronomy (Haseltine, Discover, 2002). Horace Lamb on turbulence (1932): “I am an old man now, in addition to when I die in addition to go to Heaven there are two matters on which I hope as long as enlightenment. One is quantum electrodynamics, in addition to the other is the turbulent motion of fluids. And about the as long as mer I am really rather optimistic.” Goldstein, S., 1969. Fluid mechanics in the first half of this century, Annual Rev. Fluid Mech., 1, p. 23. This story is apocryphally repeated with Einstein, von Neumann, Heisenberg, Feynman, in addition to others.

Similarly, brittle fracture of solids is commonly encountered in everyday life, in addition to still there is no real theory explaining its properties or predicting the outcome of the simplest occurrences, like breaking a glass. It is certainly a more difficult scientific problem than turbulence, in addition to while the turbulence attracted first-class mathematicians in addition to physicists, no such interest has been shown in mathematical theory of fracture in addition to large-scale de as long as mation of solids. Brittle Fracture of Solids Seismicity model This picture represent a paradigm of the current earthquake physics. Originally, when Burridge in addition to Knopoff proposed this model in 1967, this was the first mathematical treatment of earthquake rupture, a very important development. Since then perhaps hundreds papers have been published using this model or its variants. Kagan, Y. Y., 1982. Stochastic model of earthquake fault geometry, Geophys. J. R. astr. Soc., 71, 659-691

Current seismicity physical models Dieterich, JGR, 1994; Rice in addition to Ben-Zion, Proc. Nat. Acad., 1996; Langer et al., Proc. Nat. Acad., 1996, see also review by Kanamori in addition to Brodsky, Rep. Prog. Phys., 2004 – their major paradigm: two blocks separated by a planar boundary with friction. Current seismicity physical models These models describe only one boundary between blocks, they do not account as long as a complex interaction of other block boundaries in addition to , in particular, its triple junctions. Seismic maps convincingly demonstrate that earthquakes occur mostly at boundaries of relatively rigid blocks. This is a major idea of the plate tectonic. However, if blocks are rigid, stress concentrations at other block boundaries in addition to block’s triple junctions should influence earthquake pattern at any particular boundary. Geometric strain incompatibility is ignored. Example of geometric incompatibility near fault junction. Corners A in addition to C are either converging in addition to would overlap or are diverging; this indicates that the movement cannot be realized without the change of the fault geometry (Gabrielov, A., Keilis-Borok, V., in addition to Jackson, D. D., 1996. Geometric incompatibility in a fault system, P. Natl. Acad. Sci. USA, 93, 3838-3842).

Current seismicity physical models No rigorous testing of these models is per as long as med. At the present time, numerical earthquake models have shown no predictive capability exceeding or comparable to the empirical prediction based on earthquake statistics. Confirming examples are selectively chosen data. These models have a large number of adjustable parameters, both obvious in addition to hidden, to simulate seismic activity. Math used is at least 150 years old. Modern earthquake catalogs include origin time, hypocenter location, in addition to second-rank seismic moment tensor as long as each earthquake. The tensor is symmetric, traceless, with zero determinant: hence it has only four degrees of freedom – one as long as the norm of the tensor in addition to three as long as the 3-D orientation of the earthquake focal mechanism. An earthquake occurrence is considered to be a stochastic, tensor-valued, multidimensional, point process. Earthquake Phenomenology Statistical studies of earthquake catalogs – time, size, space Catalogs are a major source of in as long as mation on earthquake occurrence. Since late 19-th century certain statistical features were established: Omori (1894) studied temporal distribution; Gutenberg & Richter (1941; 1944) – size distribution. Quantitative investigations of spatial patterns started late (Kagan & Knopoff, 1980).

Statistical studies of earthquake catalogs – moment tensor Kostrov (1974) proposed that earthquake is described by a second-rank tensor. Gilbert & Dziewonski (1975) first obtained tensor solution from seismograms. However, statistical investigations even now remained largely restricted to time-size-space regularities. Why Statistical tensor analysis requires entry to really modern mathematics. (a) Fault-plane trace on a surface. Earthquake rupture starts at the hypocenter (epicenter is the projection of a hypocenter on the Earth’s surface), in addition to propagates with velocity close to that of shear waves (2.5-3.5 km/s). (b) Double-couple source, equivalent as long as ces yield the same displacement as the extended fault rupture in a far-field. (c) Equal-area projection of quadrupole radiation patterns.

Earthquake Focal Mechanism Double-couple tensor M = M diag [1, -1, 0] has 4 degrees of freedom, since its 1st in addition to 3rd invariants are zero. The normalized tensor corresponds to a normalized quaternion q = (0, 0, 0, 1). Arbitrary double-couple source is obtained by multiplying the initial quaternion by a quaternion representing a 3-D rotation (see Kagan, GJI, 163(3), 1065-1072, 2005). Using the Harvard CMT catalog of 15,015 shallow events:

Review of results on spectral slope, b: Although there are variations, none is significant with 95%-confidence. Kagans [1999] hypothesis of uni as long as m b still st in addition to s. Relation between moment sums in addition to tectonic de as long as mation Now that we know the coupled thickness of seismogenic lithosphere in each tectonic setting, we can convert surface velocity gradients to seismic moment rates. Now that we know the frequency/magnitude distribution in each tectonic setting, we can convert seismic moment rates to earthquake rate densities at any desired magnitude. Kinematic Model Moment Rates Long-term-average (Poissonian) seismicity maps

Moment rate vs. tectonic rate Tapered Gutenberg-Richter distribution of scalar seismic moment, survival function By integrating the distribution of seismic moment we obtain relation between seismic moment rate, seismic activity rate, beta, in addition to corner moment: Kagan, GJI, 149, 731-754, 2002 Naïve summation of seismic moment If the exponent is less than 2.0, the sum of power-law distributed variables converges to a stable distribution with pdf: where is symmetry parameter, are shift in addition to width parameters, in the Gaussian distribution they are only valid parameters.

Naïve summation of seismic moment For small values of moment (M) in the G-R tapered distribution, it behaves as a pure power-law (Pareto) distribution Then median (or any quantile) is proportional to hence Zaliapin, Kagan, in addition to Schoenberg, PAGEOPH, 162(6-7), 1187-1228, 2005 Holt, W. E., Chamot-Rooke, N., Le Pichon, X., Haines, A. J., Shen-Tu, B., in addition to Ren, J., 2000. Velocity field in Asia inferred from Quaternary fault slip rates in addition to Global Positioning System observations, J. Geophys. Res., 105, 19,185-19,209.

Southern Cali as long as nia earthquakes 1800-2005 Blue – focal mechanisms determined. Orange – estimated through interpolation The Cauchy in addition to other symmetric stable distributions govern the stress caused by these defects (Zolotarev, 1986; Kagan, 1990; 1994). R in addition to om rotation of focal mechanisms is controlled by the rotational Cauchy in addition to other stable distributions. Simulation results: Distribution of distances between hypocenters N(R,t) as long as the Hauksson & Shearer (2005) catalog, using only earthquake pairs with inter-event times in the range [t, 1.25t]. Time interval t increases between 1.4 minutes (blue curve) to 2500 days (red curve). See Helmstetter, Kagan & Jackson (JGR, 2005).

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