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Yan Y. Kagan Dept. Earth and Space Sciences, UCLA, Los Angeles, CA , Earthquakes and Fractures in Solids: Why do we fail to understand them and what can be done?

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Outline 1. Fracture and turbulence -- no significant theoretical progress. 2. Deficiencies of present physical models for earthquake occurrence. 3. Phenomenology: fractal distributions of size, time, space, and focal mechanisms. 4. Fractal model of earthquake process: random stress interactions. 5. Statistical forecasting earthquakes and its testing (more tomorrow at 12:00 in room 1707).

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Two Major Unsolved Problems of Modern Science 1. Turbulent flow of fluids (Navier-Stocks equations). 2. Brittle fracture of solids. Plastic deformation of materials is an intermediate case: it behaves as a solid for short-term interaction and as a liquid for long-term interaction. Kagan, Y. Y., Seismicity: Turbulence of solids, Nonlinear Science Today, 2, 1-13.

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Navier-Stokes Equation “Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations. Although these equations were written down in the 19-th Century, our understanding 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).

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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 and 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 or 11 problems by the National Research Council's board on physics and astronomy (Haseltine, Discover, 2002).

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Horace Lamb on turbulence (1932): "I am an old man now, and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics, and the other is the turbulent motion of fluids. And about the former I am really rather optimistic." Goldstein, S., 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, and others.

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Similarly, brittle fracture of solids is commonly encountered in everyday life, and 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, and while the turbulence attracted first-class mathematicians and physicists, no such interest has been shown in mathematical theory of fracture and large-scale deformation of solids. Brittle Fracture of Solids

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Seismicity model This picture represent a paradigm of the current earthquake physics. Originally, when Burridge and 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.

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Kagan, Y. Y., Stochastic model of earthquake fault geometry, Geophys. J. R. astr. Soc., 71,

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Current seismicity physical models Dieterich, JGR, 1994; Rice and Ben-Zion, Proc. Nat. Acad., 1996; Langer et al., Proc. Nat. Acad., 1996, see also review by Kanamori and Brodsky, Rep. Prog. Phys., their major paradigm: two blocks separated by a planar boundary with friction.

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Current seismicity physical models These models describe only one boundary between blocks, they do not account for a complex interaction of other block boundaries and, 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 and block's triple junctions should influence earthquake pattern at any particular boundary. Geometric strain incompatibility is ignored.

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Example of geometric incompatibility near fault junction. Corners A and C are either converging and 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., and Jackson, D. D., Geometric incompatibility in a fault system, P. Natl. Acad. Sci. USA, 93, ).

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Current seismicity physical models No rigorous testing of these models is performed. 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 and hidden, to simulate seismic activity. Math used is at least 150 years old.

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Modern earthquake catalogs include origin time, hypocenter location, and second-rank seismic moment tensor for each earthquake. The tensor is symmetric, traceless, with zero determinant: hence it has only four degrees of freedom -- one for the norm of the tensor and three for 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

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Statistical studies of earthquake catalogs -- time, size, space Catalogs are a major source of information 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).

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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.

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(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), and propagates with velocity close to that of shear waves ( km/s). (b) Double-couple source, equivalent forces yield the same displacement as the extended fault rupture in a far-field. (c) Equal-area projection of quadrupole radiation patterns.

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Earthquake Focal Mechanism Double-couple tensor M = M diag [1, -1, 0] has 4 degrees of freedom, since its 1st and 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), , 2005).

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Using the Harvard CMT catalog of 15,015 shallow events:

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Review of results on spectral slope, Although there are variations, none is significant with 95%-confidence. Kagan’s [1999] hypothesis of uniform still stands.

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Relation between moment sums and tectonic deformation 1.Now that we know the coupled thickness of seismogenic lithosphere in each tectonic setting, we can convert surface velocity gradients to seismic moment rates. 2.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

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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, and corner moment: Kagan, GJI, 149, , 2002

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Na ï ve 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: whereis symmetry parameter, are shift and width parameters, in the Gaussian distribution they are only valid parameters.

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Na ï ve 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, and Schoenberg, PAGEOPH, 162(6-7), , 2005

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Holt, W. E., Chamot-Rooke, N., Le Pichon, X., Haines, A. J., Shen-Tu, B., and Ren, J., Velocity field in Asia inferred from Quaternary fault slip rates and Global Positioning System observations, J. Geophys. Res., 105, 19,185-19,209.

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Sumatra M 9.1 earthquake

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Temporal Earthquake Distribution Omori's (1894) law: Time shift c-coefficient is the result of overlapping seismic records after large earthquake and its strong aftershocks. Singularity at t=0 means that earthquake is a cluster of events, these events resolution depends on quality of seismographic network and interpretation technique -- there is no individual earthquake!

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Spatial Distribution of Earthquakes We measure distances between pairs, triplets, and quadruplets of events. The distribution of distances, triangle areas, and tetrahedron volumes turns out to be fractal, i.e., power-law. The power-law exponent depends on catalog length, location errors, depth distribution of earthquakes. All this makes statistical analysis difficult.

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Spatial moments: Two- Three- and Four-point functions; Distribution of distances (D), surface areas (S), and volumes (V) of point simplexes is studied. The probabilities are approximately 1/D, 1/S, and 1/V.

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New ms --

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Kagan, Y. Y., Correlations of earthquake focal mechanisms, Geophys. J. Int., 110, Upper picture -- distance 0-50 km. Lower picture -- distance km. Upper solid line -- Cauchy distribution; Dashed line - random rotation.

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Kagan, Y. Y., Temporal correlations of earthquake focal mechanisms, Geophys. J. Int., 143,

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Branching model for dislocations (Kagan and Knopoff, JGR,1981; Kagan, GJRAS, 1982) Predates use of self-exciting, ETAS models which also have branching structure. A more complex model, exists on more fundamental level. Continuum-state critical branching random walk in T x R 3 x SO(3). Many unresolved claims, mathematical issues: is the synthetic earthquake set scale-invariant?

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Critical branching process -- genealogical tree of simulations

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(a) Pareto distribution of time intervals time^(1-u) (b) Rotation of focal mechanisms follows a Cauchy distribution

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Simulated source-time functions and seismograms for shallow earthquake sources. The upper trace is a synthetic cumulative source-time function. The middle plot is a theoretical seismogram, and the lower trace is a convolution of the derivative of source-time function with the theoretical seismogram. Kagan, Y. Y., and Knopoff, L., Stochastic synthesis of earthquake catalogs, J. Geophys. Res., 86,

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Kagan, Y. Y., and Knopoff, L., Random stress and earthquake statistics: Time dependence, Geophys. J. R. astr. Soc., 88,

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Snapshots of fault propagation. Rotation of focal mechanisms is modeled by the Cauchy distribution. Integers in the frames # indicate the numbers of elementary events to which these frames correspond. Frames show the development of an earthquake sequence.

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Normalized quaternions represent SO(3) group of 3-D rotations, their multiplication is non-commutative Non-commutability of 3-D rotations presents a major difficulty in creating probabilistic theory of earthquake rupture propagation.

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A model of random defect interaction in a critical stress environment explains most of the available empirical statistical results. Omori's law is a consequence of a Brownian motion-like behavior of random stress due to defect dynamics. The evolution and self-organization of defects in the rock medium are responsible for the fractal spatial patterns of earthquake faults (Zolotarev, 1986; Kagan, 1990; 1994). Simulation results:

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Earthquake Probability Forecasting The fractal dimension of earthquake process is lower than the embedding dimension: Time – 0.5 in 1D Space – 2.2 in 3D Focal mechanisms – Cauchy distribution This allows us to forecast probability of earthquake occurrence – specify regions of high probability, use temporal clustering for evaluating possibility of new event and predict its focal mechanism.

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Forecast example: displayed earthquakes occurred after smoothed seismicity forecast was calculated. Forecast effectiveness can be evaluated by the likelihood method (Kagan and Jackson, GJI, 143, , 2000).

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Time history of long-term and short- term forecast for a point at latitude N., E. northwest of Honshu Island, Japan. Blue line is the long- term forecast; red line is the short- term forecast (Jackson and Kagan, SRL, 70, , 1999).

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Kagan, Y. Y., and Knopoff, L., A stochastic model of earthquake occurrence, Proc. 8-th Int. Conf. Earthq. Eng., 1,

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WHY DOES THEORETICAL PHYSICS FAIL TO EXPLAIN AND PREDICT EARTHQUAKE OCCURRENCE? 1. There are major, perhaps fundamental difficulties in creating a comprehensive physical/mathematical theory of brittle fracture and earthquake rupture process. 2. However, the development of quantitative models of earthquake occurrence needed to evaluate probabilistic seismic hazard is within our reach. 3. It will require a combined effort of Earth scientists, physicists, statisticians, as well as pure and applied mathematicians.

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End Thank you

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Conclusions The major theoretical challenge in describing earthquake occurrence is to create scale-invariant models of stochastic processes, and to describe geometrical/topological and group-theoretical properties of stochastic fractal tensor-valued fields (stress/strain, earthquake focal mechanisms). It needs to be done in order to connect phenomenological statistical results and attempts of earthquake occurrence modeling with a non-linear theory appropriate for large deformations. The statistical results can also be used to evaluate seismic hazard and to reprocess earthquake catalog data in order to decrease their uncertainties.

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Observational results: (1) Earthquake size distribution is a power-law (Gutenberg- Richter) with an exponential tail. The power-law exponent has a universal value for all earthquakes. The maximum (corner) magnitude values are determined for major tectonic provinces. (2) The temporal fractal pattern is power-law decay of the rate of the aftershock and foreshock occurrence (Omori's law). Power-law time pattern can be extended to small time intervals explaining the complex structure of the earthquake rupture process. (3) Spatial distribution of earthquakes is fractal; the correlation dimension of earthquake hypocenters is about 2.2 for shallow earthquakes. (4) Disorientation of earthquake focal mechanisms is approximated by the rotational 3-D Cauchy distribution. Earthquake process exhibits scale-invariant, fractal properties:

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Southern California earthquakes Blue -- focal mechanisms determined. Orange -- estimated through interpolation

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The Cauchy and other symmetric stable distributions govern the stress caused by these defects (Zolotarev, 1986; Kagan, 1990; 1994). Random rotation of focal mechanisms is controlled by the rotational Cauchy and other stable distributions. Simulation results:

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Distribution of distances between hypocenters N(R,t) for 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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