Yan Y. Kagan

Dept. Earth and Space Sciences, UCLA, Los Angeles,

CA 90095-1567, ykagan@ucla.edu,

http://scec.ess.ucla.edu/ykagan.html

Earthquakes and Fractures in Solids:

Why do we fail to understand them and

what can be done?

http://scec.ess.ucla.edu/~ykagan/india_index.html

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).

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., 1992. Seismicity: Turbulence of solids,

Nonlinear Science Today, 2, 1-13.

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).

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, seehttp://feynman.physics.lsa.umich.edu/strings2000/millennium.html

or 11 problems by the National Research Council's board on

physics and astronomy (Haseltine, Discover, 2002).

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., 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, and others.

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

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.

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 and Ben-Zion,

Proc. Nat. Acad., 1996; Langer et al., Proc.

Nat. Acad., 1996, see also review by Kanamori

and 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 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.

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.,

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

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.

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

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).

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), and

propagates with velocity

close to that of shear waves

(2.5--3.5 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.

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), 1065-1072,

2005).

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

]/)exp[()/()( ctt MMMMMM −−−−====βΦ

Review of results on spectral slope, β:β:β:β:

Although there are variations, none is significant with 95%-confidence.

Kagan’s [1999] hypothesis of uniform ββββ still stands.

Relation between moment sums

and tectonic deformation1. 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

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:

)1/()2(1

00 βββα ββ −−−−−−−−====−−−−

••••

Γcs MMM

]/)exp[()/()( ctt MMMMMM −−−−====β

Φ

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

β−−−−∝∝∝∝MM )(Φ

),,,,( σµγβφ M

converges to a stable distribution with pdf:

where γ is symmetry parameter, σµ, are shift

and 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

β)/()( MMM t====Φ

Then median (or any quantile) is proportional to

βµ /1)( NN ∝∝∝∝ hence )20(8.2)40( µµ ××××≈≈≈≈

Zaliapin, Kagan, and Schoenberg, PAGEOPH,

162(6-7), 1187-1228, 2005

Holt, W. E., Chamot-Rooke, N., Le Pichon, X., Haines, A. J.,

Shen-Tu, B., and Ren, J., 2000. Velocity field in Asia inferred

from Quaternary fault slip rates and Global Positioning

System observations, J. Geophys. Res., 105, 19,185-19,209.

tectonicseismic MM••••••••

==== /χ

Sumatra M 9.1 earthquake

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!

1)()( −−−−++++∝∝∝∝ cttn

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.

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.

New ms -- http://scec.ess.ucla.edu/~ykagan/p2rev_index.html

Kagan, Y. Y., 1992.

Correlations of

earthquake focal

mechanisms,

Geophys. J. Int., 110,

305-320.

• Upper picture --

distance 0-50 km.

• Lower picture --

distance 400-500 km.

Upper solid line --

Cauchy distribution;

Dashed line - random

rotation.

Kagan, Y. Y., 2000. Temporal correlations of earthquake focal

mechanisms, Geophys. J. Int., 143, 881-897.

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 R3 x SO(3).

• Many unresolved claims, mathematical issues:

is the synthetic earthquake set scale-invariant?

Critical

branching

process --

genealogical

tree of

simulations

(a) Pareto

distribution

of time

intervals

time^(1-u)

(b) Rotation of

focal

mechanisms

follows a

Cauchy

distribution

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., 1981. Stochastic synthesis of

earthquake catalogs, J. Geophys. Res., 86, 2853-2862.

Kagan, Y. Y.,

and Knopoff,

L., 1987.

Random

stress and

earthquake

statistics:

Time

dependence,

Geophys. J. R.

astr. Soc., 88,

723-731.

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.

Normalized quaternions represent SO(3)

group of 3-D rotations, their multiplication

is non-commutative

1221 qqqq ××××≠≠≠≠××××

Non-commutability of 3-D rotations presents a

major difficulty in creating probabilistic theory of

earthquake rupture propagation.

• 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:

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.

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, 438-453,

2000).

Time history

of long-term

and short-

term forecast

for a point at

latitude

39.47 N.,

143.54 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, 393-403,

1999).

Kagan, Y. Y., and Knopoff, L., 1984. A stochastic

model of earthquake occurrence, Proc. 8-th Int.

Conf. Earthq. Eng., 1, 295-302.

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.

End

Thank you

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.

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:

Southern California earthquakes

1800-2005Blue -- focal

mechanisms

determined.

Orange --

estimated

through

interpolation

• 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:

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).